Self-replicating proteins can happen by a chance. And if the chance is too small for the life to happen accidentally in the known universe, then this is an argument that there are more than we can see or there were many universes in past.
The simplest self-replicating system we know of that exists in some natural environment (not, say, in a carefully purified solution of energetic monomers) has billions of atoms. And self-reproducing proteins have never, to my knowledge, ever been demonstrated in any experiment.
If it turned out that going from amino acid to DNA/protein life is only possible by chance, then this an argument that there were at least 1e1000000 or whatever number past universes to allow to produce life.
It's not a lottery. It's a hill-climb.
Once we have a system capable of Darwinian evolution, yes. The problem is getting to such a system. This is the Great Handwave we see so often in discussions of origin of life.
Is it really a handwave to say we don't know? Sound pretty logical as opposed to just making something up or parroting some origin story like people have been doing for 1000s of years.
It's a handwave to say it's a hill climb when there's no clear mechanism to make it a hill climb. That's not saying we don't know, it's saying we DO know when we actually don't. It's basically assuming away the problem, the complexity gap between simple chemicals and systems we know can pile on complexity via natural selection.
I wonder whether having a better understanding of the mechanism of prion replication might help illuminate possible pathways to the emergence of life.