I look forward to grinding this game more but the NES controller is really hurting my thumbs. There is a controller by Retro Fighters called the Jab for the NES. I wanted to get it because it has analog sticks as well as the button layout being better ergonomically. Would this be ok to use for speed runs? There is no unfair advantage.

Yeah, that's ok!

As long as the controller has no advantages or functions that can't be replicated on real controller (example - turbo, L+R inputs) then controller doesn't matter.