I thought it was implied that he didn't want the responsibility because the gov't/bureaucracy sucks ass in the setting?
This, I don't know why people are shitting him so much when he's made it clear that he doesn't want to do a bunch of extra work for basically no reason. He may lack ambition but a lack of ambition is not necessarily a negative quality when you are not wanting for anything and are perfectly happy with your current situation.
The lack of respect sucks but he knows the only way he'd get respect is to put in the effort and work only for him to get better social standing (which he doesn't care about). The people who are wrong are the ones who put too much weight on social standing above just being a decent person. That constant drive to one up everyone else and prove yourself superior is not healthy and prevents you from forming meaningful relationships as everyone becomes either a stepping stone for you, a competitor, or worse a trophy.
Let's analyze the other people in this story. The main antagonist is a spiteful wretched person who doesn't ever seem to be happy and constantly shits on others in order to reinforce his superiority because he has nothing beyond that. Even among the students the only reason people flock to him is because of his rank and fame, not because he's enjoyable to be around (people even specifically hate his personality but they begrudgingly follow him because they also want that fame by association). Meanwhile Washio is content and just enjoys teaching and now has some big titty monster girls fawning all over him. Not because he's famous but because when he started out in this world he did something no one else bothered to, teach a bunch of monsters. It also begs the question of 'Monster Girls' becoming enlightened specifically due to his guidance which would make his efforts akin to a god maker considering how they are idolized.
In short, don't be an overly ambitious asshole who throws decency and humanity aside for power and status because it doesn't get you anywhere except an empty life with no true interpersonal relationships. Be considerate and actually care about the people around you and maybe you too will get big titty monster girls to want to jump on your dick (metaphorical or literal).