So I think he thought his job of protecting him was done. Because once he takes the black, he's supposed to lose any inheritance claims (which he never would've had) and wouldn't have been in a position where he could've threatened Robert. I'm guessing that's why Robert never went after Aemon Targaryen. By joining the Night's Watch, Jon was unknowingly removing himself from ever being a threat to Robert even if he did find out the truth and would no longer be seen as a threat to his legitimacy.

Or at the very least, once he joins the Night's Watch, he'll be trained to take care of himself and have a whole brotherhood looking out for him. Kinda like how some parents will treat their job of raising their kids as "done" when they graduate high school.