If there were some proof that this universe was the only one that exists, wouldn't that infidels analogy to a lottery fail? In a lottery there are many losers, but there will definitely be a winner. If there was only one universe that was randomly assigned constants, and there didn't have to be a winner (in fact, it was astronomically implausible that there would be a winner), I could see design as the best inference to why we are around to think about it
Forgot to reply, my bad.
The analogy is attacking the following intuition: "If X is extremely unlikely then we are justified in thinking that X was designed."
It's also worth noting that the above is a false dichotomy. Chance or design are not exhaustive as there could exist another option such as logical necessity.
The intuition behind this defence of the fine tuning argument seems to rely on the fallacy of false cause.
Take the following example:
(1) A single lottery, chances of me winning are 1 in 1,000,000,000
(2). Three lotteries. The chances of me winning each individual lottery is 1 in 1,000. Chances of me winning all three is 1 in 1000^3
If we don't assume that there has to be a winner, both (1) and (2) have the same likelihood as they share the same mathematical probabilities.
If we could prove that one universe is all that there is, then the multiverse objection would obviously break down. The actual challenge lies in defending the claim that this universe is the only one. However, I don't think that this discovery would change the fact that the "chance or design" dichotomy, is a false one.
I just want to add that I think the multiverse objection is one of the weaker responses to FT. Imo, there are stronger philosophical arguments against teleological fine tuning.