One could argue that PSR violations are generally rare, fleeting, and minor so we would naturally have no experience with them. This isn't my position. But it would be a coherent one to take.
I think there are at least three potential problems with making this sort of move. Apologies for the incoming autism:
1. Let H1 refer to the unrestricted PSR (i.e., all contingent facts are explicable) and let H2 refer to a restricted PSR (e.g., m/n contingent facts are explicable, where m/n < 1). In the "odds" form of Bayes' theorem, we have: P(H1|E) / P(H2|E) = [P(H1)P(E|H1)] / [P(H2)P(E|H2)]. Assuming that P(H1) = P(H2)—which is being generous, see below—the relevant comparison is between P(E|H1) and P(E|H2). It seems rather obvious that the former is greater than the latter, even if only to some small degree, and so the total body of evidence supports the unrestricted PSR over the restricted PSR.
2. How is the restricted PSR epistemically justified? It’s clearly not justified a priori given its form (compare: "90% of trucks are red" isn’t the sort of proposition that could be justified a priori). So is it empirically justified? The problem with this option is that any experiences or purported observations we appeal to are the sorts of things that are, for all we know (assuming the falsity of the PSR), snapping into existence without explanation. Thus, it doesn’t seem we could claim it’s empirically justified either.
3. Most fundamentally, it’s not clear how we can assign an objective probability to some inexplicable contingent state of affairs. Objective probabilities depend on the objective tendencies of things, but if the PSR is false, events can occur in a manner that has nothing to do with objective tendencies. In other words, regarding contingent facts, it seems objective probabilities are downstream of explicability. You might argue that we can, in principle if not in practice, simply count all the inexplicable facts and compare them to the explicable facts to determine a probability. However, that would presuppose the classical theory of probability, which is rejected by virtually every probability theorist today.
I wanted to keep these initial remarks relatively brief, but I can expand on any or all of them if you're interested.
The issue here is that you're presuming some sort of "mind" to whatever necessarily existing "thing" gave rise to the rest of the universe.
It's not a presumption; it's an argument: If realism concerning abstract objects (universals, propositions, possible worlds, etc.) is true, then either Platonism is true or theism is true. Realism is true. Platonism is false. Therefore, theism is true.
To be clear, I'm not defending this argument here; I was only offering it as an answer to the "What does this have to do with theism?" question.