I'd say that all of "Universe appeared out of nowhere", "Prime mover appeared out of nowhere, created universe", and "Prime mover has always existed, created universe" are incredibly unlikely. "Universe has always existed" doesn't work; even if a neverending cycle of big bangs are possible, they had to start somewhere.
Invoking the anthropic principle, perhaps wrongly, I assert that NONexistence is the correct status quo, thus existence is unusual, and needs an explanation one way or the other.
Secondly, with all the billions of things you have to shotgun at, you'd expect at least one of them to give a staggeringly unlikely result.
Thirdly, there could easily be a mechanism explaining it. Wow, in a right angled triangle, a^2 + b^2 = c^2 is always true! It's a huge coincidence, right?
As a side note... would you count the fact that we have not found anything of this nature as evidence that a deity doesn't exist?
To answer that 'secondly': Depends. If we've got "billions" of things to shotgun at, and we find something that appears to be the product of an intelligence that has not a one-in-a-billion chance, but one-in-a-billion-billion-billion-billion chance, I still argue that that would be solid evidence.
To answer that 'thirdly': Yes, finding the mechanisms that caused an incredibly unlikely result does defeat their use as evidence. However, I would argue that the very existence of some of these mechanisms are, themselves, evidence of a creator. I acknowledge that
many, or even most, would disagree with me here. I assert that chaos is the status quo, and the order we observe is unusual. The fact that everything fits together so neatly inspires faith in me, and in some others. I'm not offended by others who assert that order is the natural status quo, though. I think that stuff like this is a very valid difference of opinion, but I've seen people get nudged closer to religion just by studying math.
To answer that side note: Yes. I do count the fact that we haven't found anything of that nature as evidence against the existence of a creator. I still feel that the evidence
for is greater.