Yes, in the long term, the planet will be fine. But bear in mind, our entire biology is based on converting O2 into CO2.
I mean, sure, a couple billion years ago, the global ecosystem had the opposite problem and single-celled archaea was suffocating the planet with too much O2. Those are the conditions that allowed animal life to evolve.
So, I take your point that the planet will still have O2 long after we flood the atmosphere with the millions of tons of CO2 that used to be buried deep underground. Plankton will have a comeback even if the vast majority of animal life on the planet dies of asphyxiation first. But at that point, the argument of whether we’ve “run out” of O2 is really semantics, right? If we haven’t “run out” of it, but our supply gets low enough that virtually all of us are dead as a result, I don’t place a lot of value in making that distinction.