The issue with the challenges that occurred prior to the election, and between the election and the inauguration, is that they had the wrong plaintiffs. I believe, based on this opinion, that the only real chance of getting a full hearing on a candidate’s qualifications would be for another candidate (major or minor) to bring a challenge prior to the election (or, as you mentioned, maybe in between the election and the inauguration).
Standing is a tough hurdle to clear, but it is an important check on judicial power.
Or a means of judicial tyranny. Take your pick.