Actually, I believe it's worse than that. It's the House of Representatives that gets to decide, when the electoral votes are officially counted.
If that’s true then we have a hole in our constitution that any foreign billionaire could drive a truck through. Let’s say some 30 year old saudi billionaire jerkface dumps $2B into an election, gets elected and spreads all kinds of payola around to congress. Suddenly that 35 year old requirement for eligibility is toast.
Our founding fathers were concerned that someone with divided loyalties would angle their way through the system and that is the source of this “natural born” requirement.