No. The mapping of nucleotide sequences onto amino acids isn't predetermined. We've built in the labs versions that are different.
OTOH, the argument still isn't good. It could be a low, but not extremely low, probability. In that case the first one to show up could have a VERY strong advantage. And we haven't checked all life on Earth, so the assertion that they are all the same hasn't really been proven, either.
We are pretty certain that the appearance of life involved some very low probability events, but that there were a lot of environments around with lots of different samples for a long period of time, so a "low probability event" should be expected to show up (even if not any particular low probability event).