I don't think the claim requires a uniform distribution, just that the values come from some possible distribution (of any shape). With enough (or infinite) shots on goal, you're gonna get all combinations of them.
The question "why these values of constants instead of others?" sort of presupposes that other values are possible. If you instead believe that the values are fixed, then your answer is just "because that's the only value that's possible."
Yes, you're right, the distribution need not be uniform. The assumption is that the distribution is such that the probability that all the constants are within the Goldilocks range is very low (not even necessarily that the constants are fixed). But yes, either way, my answer is that there's no reason to believe this assumption.
