I still don't understand what it means to claim that a president is "illegitimate."
It means whatever you want it to mean. Eye of the beholder.
I can think of two general analogies: the Borgia popes and the antipopes. The leaders of the House of Borgia were suspected of adultery, incest, simony, theft, bribery, and murder, among other things, and they clawed their way to the top using trickery and broken promises. Antipopes simply oppose the legitimately elected pope and made claims to be actual pope. Trump is more like the Borgia popes than the antipopes, I think, in the sense that no one has decided to oppose his presidency and declare himself or herself the actual president, and in the sense that Trump engaged in a general sort of tomfoolery to get himself elected, and again to pursue a legislative agenda.
I wouldn't call him illegitimate, but then I wouldn't call the Borgia Popes illegitimate either. Powerhungry, perhaps, and unethical, but not illegitimate. Still, I can understand that logic that might lead others to conclude that a certain illegitimacy obscures their moral authority.
