> but that doesn't mean it's been proven through experimentation, does it? isn't it just an assumption?
It’s not an assumption, it’s backed up by excellent evidence — see the paper I linked.
> what if there's something akin to mathematical uniqueness in the mechanics of the core machinery?
Well that’s clearly not the case, we can trivially (…) design self-replicating machines that have completely different mechanics, as a thought experiment. More to the point, we can change parts of the machinery. For instance, we can take the universal genetic code and, with effort, change it into something completely different (by just swapping all codons around). The result is just as viable, but doesn’t exist in nature. In fact, the observed universality of the genetic code, in itself, is already seen as sufficient evidence for common descent (and then some).