I'm currently running on a gamepad (8bitdo SN-30+) but I have an arcade stick with a quad gate joystick and I've been debating switching to it because 1) it's easier on my thumb and 2) it reduces the number of inadvertent diagonal inputs... but I'm wondering if that would be considered an unfair (NES) advantage?

I see no problems with it. As long as it does not give you access to functions unavailable to players on regular controllers (30Hz mashing e.g.) it's fine to use imo.