"Sure, there would be an awful lot of them, but infinity? No."
Maybe, maybe not. Was the Big Bang the start of the universe or merely a local event in an infinite preexisting universe ? Could have had infinite time & space to have infinite universe states. And while splitting is part of the initial many worlds theory, I suspect the inability to describe how that occurs, and where the energy to do it comes from, possibly makes an 'already infinite states, no splitting needed', one more likely.