Are there any issues with this syllogism as regards a natural origin of life (abiogenesis)?

I would remove "abiogenesis" and stick with terms in the premises. The premises seem to suggest that cellular life is infinite. If the goal is to prove "then God did it," this would also make impossible the God hypothesis (i.e., God is alive and he requires proteins, but since P2 - God is impossible).

Also fact-checking the premises you will see:

Most protein synthesis does happen inside living cells on ribosomes. But there are clear counterexamples: 1. Cell-free translation: You can make proteins in a test tube using purified translation machinery or extracts (e.g., E. coli lysate, wheat-germ, rabbit reticulocyte, the PURE system). No living cells involved. 2. Chemical synthesis: Solid-phase peptide synthesis and native chemical ligation can build whole proteins (insulin, ubiquitin, small enzymes) entirely without cells or ribosomes.
3. Abiotic peptide formation: Lab models of prebiotic chemistry can form short peptides on mineral surfaces or via wet–dry cycles—again, outside life.

A more accurate claim: All known biological, coded protein synthesis uses ribosomes that originate from life, and in nature it occurs inside cells. But protein synthesis itself is not restricted to living cells.

You may want to work this into the rewrite.