Does the point in time when life emerged suggest it as a common occurrence?

The reason we believe that all life on Earth belongs to the same lineage is that every life form we’ve ever looked at has the exact same biochemical basis. Not just amino acids and nucleic acids and lipids in some form, but all life forms use the exact same genetic codes for the exact same amino acids.

If a brand new life form sprang forth from the primordial ooze, would we expect that brand new life form to use Uracil-Uracil-Uracil in it’s triplet codons to specify Phenylalanine, and Adenine-Adenine-Adenine to specify Lysine, and Adenine-Adenine-Guanine to also specify Lysine, but Adenine-Adenine-Cytosine to specify Asparagine?

The entire triplet codon table seems completely arbitrary, yet every single life form on Earth uses the exact same coding scheme. Every one of the 20 amino acids used by every single life form on Earth have the exact same chirality, yet when produced inorganically those amino acids are a 50-50 mix of chiralities. There’s no particular reason that we only use the 20 amino acids we do, there’s no particular reason those amino acids have to have the chirality they do. Yet those features are exactly the same in every life form on Earth.

Even if something in the nature of life neccessitated nucleic acids that coded for amino acids that assembled into proteins surrounded by lipid membranes, it’s beyond belief that the exact same assemblage of exactly identical chemicals assembled in exactly the same way more than once.

There is a possibility that present-day life is an amalgamation of independently arising lineages of proto-life, or that there were competing lineages early in Earth’s history. But all extant life that we know of has the same origin. That doesn’t rule out the possibility that tomorrow we’ll find an organism that doesn’t share this lineage, just that since we never have yet despite looking at thousands and thousands of species it seems pretty unlikely that the next bacteria we look at will turn out to the the oddball.

And even if we do turn up such an oddball there’s no saying it arose on Earth–it might have come from an extra-terrestrial biosphere. Of course, so could life on Earth. Maybe it would be the only remnant of the original Earth biota, and all the other life on Earth is from an external source.