Short answer is that birtherism lets you outright disregard Obama as legitimate. The fact that he's black/the other ties into that.
It is also easier to sell to the idiot masses who are already triggered by a black man with a foreign name being President.
Also, the Russian stuff is full-tilt revenge not only for the birtherism stuff but also for the eight years the Democrats were taking it up the ass and then some under Bush, as far as an entire generation of Democrat operatives (and those still around) still having PTSD over remembering how powerless and castrated they were under Bush after 9-11 gave him immunity from criticism and +50 charisma in terms of evoking 9-11 to get his way. And if you want to go further, for all of the shit Clinton suffered in his Presidency, as far as how there was a huge cottage industry set up on the right looking for ways to fuck Clinton over/get him out of office by any means necessary.
The irony is that Trump doesn't need ALL of the Black Vote.
All he really has to do is peel off just enough of the black vote to butt-fuck the left, since all of their political math these days involves strategies that factor in getting ALL of the women, minorities, and gay votes to form a superblock to give them a super majority. A strategy that is utterly exceptional in that it explicitly requires ALL of those group voting Democrat and don't fucking factoring in ANY portion of them saying "fuck you" and voting Republican.
He just needs to get enough votes from black people that he can kneecap the Democrats with their demographic math scheme, because the Democrats math falls apart if ANY portion of the black vote (or other minorities or women and gays) goes the other way and votes GOP.