Booth was a bit of a delusional wack job, he thought he would be welcomed like a hero in the south and would be protected like a folk hero.
If Mudd had moved his family to VA ahead of time that would be a smoking gun.
Think about not having anyway to call the cops, miles away from the nearest neighbor and an unhinged guy that you might be afraid of shows up, I'd imagine the easiest way Mudd thought to get rid of him was to give him what he wanted.
Lincoln's death led to the south being punished, where he wanted a policy of malice toward none. So thanks to Booth the south was treated much like the left wants people "cancelled" so the left should be happy Lincoln was assassinated.