Cross-My-Primes

Hello Math-Fun,
this game (hope not old hat) is played on a kind of infinite Scrabble board where all squares are white, and no square has any text on it. The starting "star" square, in the center of the grid, has a 7 on it (the number, not the word, as this game is played with digits only that will form numbers – instead of letters forming words). 

The single player must now form prime numbers at every turn on the board – at the first turn placing the integer 1, at the second turn placing 2, then 3, then 4, then 5, etc. – this is the natural order of the positive integers as they appear (and because 7 starts the game, the player will jump from 6 to 8. No other jump will be allowed during the game). 

Those integers, as in the traditional game of Scrabble, must be attached to the existing structure at least by a digit (one digit per square). 

All visible numbers, before and after any turn, must be prime: they are read horizontally from left to right or vertically from top to bottom (again, like words in a traditional Scrabble game). 

Example here:

We start the game with 7 on the star:
....................7....................

then comes 1 and we form the prime 71;
....................71....................

then comes 2 and we form the prime 271;
...................271....................

then comes 3 and we form the prime 3271;
..................3271....................

then comes 4 and we form the prime 43271;
.................43271....................

then comes 5 and we form the prime 53 (5 on top of 3 on the grid);
..................5.......................

.................43271....................

then comes 6 and we form the prime 61 (6 on top of 1 on the grid);
..................5..6.....................

.................43271.....................

then comes 8 (because 7 is already on the board) and we form the prime 853 (8 on top of 53 on the grid);
..................8........................
..................5..6.....................

.................43271.....................

then comes 9 and we form the prime 29 (9 under 2 on the grid);
..................8........................
..................5..6.....................

.................43271.....................

...................9.......................

then comes 10 and we form the prime 1061 (10 on top of 61 on the grid);
.....................1.....................

..................8..0.....................

..................5..6.....................

.................43271.....................

...................9.......................

now comes 11 – we have many choices here; we can place vertically 11 sticking tot he upper digit 1 of 1061 to form twice the prime 11 (horizontally and vertically) :
......................1....................
.....................11....................

..................8..0.....................

..................5..6.....................

.................43271.....................

...................9.......................

Etc.
After a few discussions with friends, we've decided to forbid to split – as this is too much a help in building the structure. The other problem being that once a number is split, it is difficult to find it looking at a completed grid. This goes against the rules of Scrabble, we know. So, the placement of 11 hereunder is now forbidden, though 101 is a genuine prime:
...........................................
.....................1.....................

..................8.101....................

..................5..6.....................

.................43271.....................

...................9.......................

Is it possible to place, one after the other, starting with 7 in the "center" of the grid, the first 100 natural integers on it? If yes, is there a compact solution that beats the others?

It might be impossible to place the first 100 integers, I do not know – but in this case, is there another start than 7 on the "center star" that would allow that?

This suggests to start a new game every now and then with other primes than 7; one could start a new game with primes like 11, 13, 17 or 17623 (it seems to me that the game halts quickly with the starts 2, 3 and 5).

And now, perhaps, a sequence for the OEIS (any takers?):

We start we the empty board and the first prime on the "star":

...........................................

...........................................

...........................................

....................2......................

...........................................

...........................................

S = 2, ...
We always want to extend S with the smallest integer not yet present in S that can be placed on the grid, according to the above rules; a(2) will be equal to 3 as no possible prime can be built with 1 and 2:
...........................................

...........................................

...........................................

....................23.....................

...........................................

...........................................

S = 2, 3, ...
Now 1 can be placed (forming the prime 31, for instance):
...........................................

...........................................

...........................................

....................23.....................

.....................1.....................

...........................................

S = 2, 3, 1, ...
Now 4 can be placed (forming the prime 431):
...........................................

...........................................

.....................4.....................

....................23.....................

.....................1.....................

...........................................

S = 2, 3, 1, 4 ...
Now 5 can be placed (forming the prime 5431):
...........................................

.....................5.....................

.....................4.....................

....................23.....................

.....................1.....................

...........................................

S = 2, 3, 1, 4, 5, ...
Now 6 cannot be placed (on this grid)  – but 7 yes (forming the prime 75431 for instance):
...........................................

.....................7.....................

.....................5.....................

.....................4.....................

....................23.....................

.....................1.....................

...........................................

S = 2, 3, 1, 4, 5, 7, ...
Now there is a place for 6 – which will form the prime 67:
...........................................

....................67.....................

.....................5.....................

.....................4.....................

....................23.....................

.....................1.....................

...........................................

S = 2, 3, 1, 4, 5, 7, 6, ... etc.
Question:
What is the lexicographically earliest sequence of distinct positive terms fitting the "Cross-My-Primes" rules?
Best,
É.
__________________
Update of May 5th, 2020
Maximilian H. computed quickly the hereunder grid, starting on 2 and placing (without splitting them) the first 100 integers. Only 5 integers (in yellow) appear a bit later than they should – but not very late!
The sequence I was looking for is thus quite boring. Here are the first 37 terms of S (the next ones are 38, 39, 40, ... 98, 99, 100 with no "yellow inversion"):

S = 2,3,1,4,5,7,6,8,9,10,11,12,13,15,14,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,33,32,35,36,34,37,...

How was computed Maximilian's wonderful structure?
1) start with the prime 2;
2) always try to place on the grid the smallest integer not yet present;
3) any new integer must be linked to the previous structure by at least a single digit (like in the game of Scrabble);
4) always try to keep the structure compact;
5) splitting integers is forbidden.
Voilà!

[In French (merci Maximilian!) :

1° - placer toujours le plus petit nombre possible (on ne peut pas placer un plus petit, à n'importe quel instant) ; de plus :

2° - parmi ceux-là, ne considérer que ceux qui permettent le plus petit placement au coup suivant (par exemple, avec le 1 on peut faire 13 et 31, mais on retient le 31 permettant de placer le 4 au coup suivant, ce qui n'est pas possible en faisant un 13) ;

3° - parmi ceux là (minimal au coup n et aussi n+1) on choisit celui dont le "barycentre" (milieu) est le plus proche de l'origine (0,0), là où il y a le 2 bleu initial – pas du barycentre de la structure déjà produite) ;

4° - parmi ceux qui sont toujours "équivalents", on prend celui qui a les plus petits "square spiral numbers" (le max de ces nombres pour chaque case sur laquelle est placé un de ses chiffres doit être minimal).]

(The integers 1 to 9 are in red – except 2, in blue; the yellow integers must be read vertically, from top to bottom – and there is sometimes a mark to separate two of them; the underlined integers must be read horizontally, from left to right.)
Under the structure is a table showing the primes produced by the placement of the successive natural numbers – see what 41 does!
Maximilian has used a very convenient notation to locate any integer k on the grid:
2 has coordinates [0,0], 3 = [1,0], 4 = [1,-1], 5 = [-1,0] and 6 = [-1,-1].

2 forms 2	2=[0,0]
3 forms 23	3=[1,0]
1 forms 31	1=[1,1]
4 forms 431	4=[1,-1]
5 forms 523	5=[-1,0]
7 forms 7523	7=[-2,0]
6 forms 67	6=[-2,-1]
8 forms 8431	8=[1,-2]
9 forms 84319	9=[1,2]
10 forms 109	10=[-1,2]
11 forms 11 and 167	11=[-2,-2]
12 forms 12109	12=[-3,2]
13 forms 113	13=[2,1]
15 forms 157523	15=[-4,0]
14 forms 14157523	14=[-6,0]
16 forms 163	16=[3,-1]
17 forms 17 and 11	17=[-1,-3]
18 forms 1811	18=[-4,-2]
19 forms 19 and 89	19=[2,-3]
20 forms 2011	20=[-1,-5]
21 forms 11321	21=[4,1]
22 forms 222011	22=[-1,-7]
23 forms 223	23=[-2,3]
24 forms 2484319	24=[1,-4]
25 forms 2512109	25=[-5,2]
26 forms 263	26=[-4,4]
27 forms 248431927	27=[1,3]
28 forms 263287	28=[-1,4]
29 forms 229	29=[2,3]
30 forms 30263287	30=[-6,4]
31 forms 11 and 31	31=[4,-2]
33 forms 233	33=[0,-6]
32 forms 32233	32=[-3,-6]
35 forms 351811	35=[-6,-2]
36 forms 3631	36=[4,-4]
34 forms 343631	34=[4,-6]
37 forms 23, 17 and 37	37=[4,2]
38 forms 383	38=[-6,-4]
39 forms 3919 and 29	39=[2,-5]
40 forms 401	40=[-4,-4]
41 forms 41, 541 and 1671223	41=[-3,1]
42 forms 4232233	42=[-5,-6]
43 forms 167122343	43=[-2,5]
44 forms 443	44=[-4,6]
45 forms 45351811	45=[-8,-2]
46 forms 463	46=[1,-8]
47 forms 47 and 47	47=[-3,-5]
48 forms 48463	48=[1,-10]
49 forms 24843192749	49=[1,5]
50 forms 503	50=[-8,-4]
51 forms 1151	51=[5,-1]
52 forms 521	52=[6,-3]
53 forms 953	53=[3,4]
54 forms 54521	54=[6,-5]
55 forms 5545351811	55=[-10,-2]
56 forms 56443	56=[-6,6]
57 forms 557	57=[4,4]
58 forms 5814157523	58=[-8,0]
59 forms 59 and 59	59=[0,6]
60 forms 60343631	60=[4,-8]
61 forms 461 and 3147	61=[-5,-5]
62 forms 622512109	62=[-7,2]
63 forms 115163	63=[7,-1]
64 forms 643	64=[8,-3]
65 forms 65222011	65=[-1,-9]
66 forms 66347	66=[-3,-8]
67 forms 3767	67=[6,2]
68 forms 685545351811	68=[-12,-2]
69 forms 95369	69=[3,6]
70 forms 7030263287	70=[-8,4]
71 forms 71, 1777 and 61	71=[5,3]
72 forms 727 and 7622512109	72=[-8,2]
73 forms 73, 571777 and 545213	73=[5,0]
74 forms 74545213	74=[6,-7]
75 forms 755814157523	75=[-10,0]
76 forms 764232233	76=[-7,-6]
77 forms 577, 37 and 67	77=[4,5]
78 forms 78643	78=[8,-5]
79 forms 79 and 79	79=[-1,7]
80 forms 809	80=[-3,8]
81 forms 6781	81=[5,6]
82 forms 823	82=[-7,-8]
83 forms 83 and 89	83=[2,7]
84 forms 8448463	84=[1,-12]
85 forms 858448463	85=[1,-14]
86 forms 86461	86=[-5,-8]
87 forms 376787	87=[8,2]
88 forms 887	88=[9,0]
89 forms 89 and 83	89=[1,8]
90 forms 6907 and 739	90=[7,0]
91 forms 691	91=[-5,7]
92 forms 92764232233	92=[-9,-6]
93 forms 55793 and 619	93=[6,4]
94 forms 941	94=[-7,8]
95 forms 9560343631	95=[4,-10]
96 forms 9666347	96=[-3,-10]
97 forms 97, 61991 and 37	97=[6,5]
98 forms 987030263287	98=[-10,4]
99 forms 8999, 5779 and 89	99=[4,7]
100 forms 10056443	100=[-9,6]

Bravo et merci, Maximilian, quelle merveille !