The two aren't mutually exclusive.
Judo is exceptionally important to Nagatoro. To the point to where I believe that when we first meet her she is actually a little depressed and a bit lost. Think about it: the first panel that we can see her full face she looks downright depressed. She is bored of boys at 15/16. She has given up on her passion for Judo. She can't commit to a club despite being an insane athlete. She is wandering around aimlessly in life, nothing lights her world up. Naoto and his quiet passion for art lit a tiny flame in her to set her in motion, and now she is this seemingly unstoppable force, even when she doesn't see it in herself. So yes, her defeating her demon represented by Orihara is quite important to her. Of course this is just my interpretation but I like to think it is fairly well thought out.
In Japanese culture, to the extent of my knowledge, men don't often actually say "I love you" verbally, they say "I love you" by making promises. Him promising to kiss her if she wins also has that cultural element to it. Honestly I think he is saving his relationship by doing what he is doing now. I think as long as she doesn't give up on herself and gives it her all, he will kiss her anyways regardless of win or loss. But he doesn't want to take away any of her motivation or invalidate his promise. Seeing as he knows how important Judo is to her.
Taking care of your woman's emotional needs is the true Chad move.
