Who says the only natural mechanism for explaining the free parameters is "chance"? What would chance even be in an unintended formation of a universe? This appeal to chance seems to presuppose a multiverse.
If someone wins numerous lotteries, we'd think the winner is cheating because the lotteries would be independent events. The fundamental constants in the universe would apply only to the single event of our universe's formation, so the analogy with winning multiple lotteries is weak.
Also, the fine-tuning argument here is indeed an appeal to a God of the gaps, because we don't even know how many constants there are. The fundamental scientific theory is unfinished in that there's no settled explanation of quantum gravity.
See the Wikipedia page on finetuning: 'In some candidate theories, the number of independent physical constants may be as small as one. For example, the cosmological constant may be a fundamental constant, but attempts have also been made to calculate it from other constants, and according to the author of one such calculation, "the small value of the cosmological constant is telling us that a remarkably precise and totally unexpected relation exists among all the parameters of the Standard Model of particle physics, the bare cosmological constant and unknown physics."'
