Actually, it's far from clear that that would have happened if Obama had refused to step down in such a scenario - arguably, it's the inauguration that makes you President. And flaws in the process until then might not be able to nullify the swearing-in.
Arguably. IANAL, but I'm part owner of the bar nearest our university's law school and I get to talk to lawyers, law students, and law professors all the time about such things. The oath of office must be taken before the President can exercise any powers of the office, but he becomes President on noon of the appropriate day, at the expiration of his predecessor's term. Taking the oath of office is the only legal aspect of the inauguration, and the reason why it's done just before noon. The rest is just a celebration. Technically Joe Biden had to wait a few minutes after taking the oath, until noon, to actually be able to exercise the power. You can't have two Presidents at the same time, but you can have two people who have taken the oath of office. Conversely, Lyndon B. Johnson became President the instant John F. Kennedy died, but he could not exercise any powers of the office until sworn in. I'd have to concede that's a distinction without much difference.
The Congress having certified the vote of the Electoral College is the point beyond which the identity of the President-electis no longer in question. At that point, the President-elect is the only person for whom taking the oath of office would have any legally cognizable meaning. The purpose of the vote certification is to raise and resolve any questions in the vote before it becomes legally binding. Due process having been satisfied, further questions are then moot. And it is unclear that this particular action of the Congress would be reviewable by any court.
If it is discovered later that the President is or was ineligible to the office, then I believe the only means to remove him is by impeachment. I can't imagine that removal would be automatic, because evidence of his ineligibility would be subject to trial. And aside from the President's death or incapacity, the only power by which a President of sound mind and body can be removed from office is by impeachment and removal by the Congress. No court can do so. I imagine that a court can issue a declaratory judgment that the President is ineligible, but it would still require someone with the power to remove the President to act upon it.
And then, as has been copiously described, the Vice-President becomes President. The claims of any losing candidate in a prior election are extinguished when the Electoral College vote is certified.
