As such, he is entirely illegitimate (at least if you go by the constitution, which may be irrelevant at this point)
That's a blatant lie even for you.
Article II, Section 2, Clause 3, commonly known as the Recess Appointment Clause, provides that,
The President shall have Power to fill up all Vacancies that may happen during the Recess of the Senate, by granting Commissions which shall expire at the End of their next Session.
I think we all remember the Bush II era when dingy Harry Reid blocked Bush's appointments and refused to let the Senate adjourn (by holding pro forma sessions lasting a few seconds every three days) to prevent Bush making recess appointments to fill those vacancies.
When Mitch McConnell subsequently did the same to Obama (dingy Harry Reid having established the precedent, and what's sauce for the goose yada yada yada), Obama ignored the Constitution and made the appointments anyway, leading to a rare 9-0 rebuke from the Supreme Court that Obama could only make recess appointments during an actual recess
NATIONAL LABOR RELATIONS BOARD v. NOEL CANNING ET AL.
BREYER, J., delivered the opinion of the Court, in which KENNEDY, GINSBURG, SOTOMAYOR, and KAGAN, JJ., joined. SCALIA, J., filed an opinion concurring in the judgment, in which ROBERTS, C. J., and THOMAS and ALITO, JJ., joined.