Personally I've always thought it was kind of stupid for basketball coaches to wear suits while they coach.
Yeah, even more so with all the assistants.
On the other extreme you have baseball managers wearing uniforms--as if they're going to take the field.
Imagine how odd it would seem if a basketball coach wore a uniform or a baseball coach wore a suit.
Belichick wears a hoody almost every game, do you think I give a damn?
Actually, I do, the red hoody is why the Pats lost to the Giants.
Besides that, he can wear any thing he wants.