I would argue that the names we give mathematical objects are more arbitrary than the underlying structure of those objects. You wouldn't say that any of mathematics would break if we renamed 2 to "smorf," or if we called addition "kupo-ing." By that metric it seems disingenuous to say that any theorems would break if prime numbers included 1. Personally, I don't think it's "far, far" simpler to leave it out, as leaving it in would amount to adding a single word to the statement of most theorems involving prime numbers. It is definitely cleaner to leave it out, and I doubt anyone would argue with that, but the distinction isn't particularly important.
