Crankin, no, the disc brakes have nothing to do with how the bike shifts. MTB trigger shifters/derailleur combos vary tremendously on how they easily and fast they shift. Be sure to position and adjust the trigger shifters so they're comfy to reach when your hands are on the grips. I like to have my thumb pad resting gently on the trigger(s) as I ride. That can make quite a difference. Same for the brake levers. Move them as needed to be comfy. On flat bar brake levers, there's usually a reach adjustment as far as how far in the lever has to go for the brakes to engage. I always have to make these adjustment when I bring a new bike home.