Help with a puzzle that is a bit too complex

[TheGreatGoatsbie]

I just tried to understand the rules. Some points are not clear to me:

Is it allowed that two primes share a digit? E.g. the 3 in R4C4 is contained in instances of 13, 23, 31 and 37.

And if two pcn share a digit, do they touch orthogonally?

Do the rules apply to each instance of a prime?
The pcn 53 in R4C2/R5C1 touches the pcn 2 in R4C1.

Good questions. First of all I should have clarified that the primes may not overlap, at all. While some prime numbers appear double in the grid, there is only one configuration that allows all prime numbers to appear, except 11.

But the last point is one I should have thought about. This is ofcourse the case for all single digit primes, of which 5 and 2 violate my current rule multiple times.

Let's say I want to place one set of all the primes in the grid on the diagonals. (left to right seems easier to me, but I'm not sure it is) For that set of primes, pcn's may never touch eachother(!) orthogonally, there are no overlapping primes. e.g. r2c2 - r3c3 - r4c4 is no valid instance of the primes 31 and 13, only one of them can count for this puzzle.

In other words, for every prime number, at least one of the pcn's has to appear in the grid that is not touching wor overlapping with another pcn, whereas at least one of every npcn has to appear in the grid.

So while I love the concept, I cannot really find a proper way to make it a non-laborious exercise. I think it is too much to keep track of the digits contained within the primes.