Your situation sounds pretty normal!
I agree with taking an empathetic path that helps you know more about your son than just insisting on certain extracurricular activites. And I guess being alert to whether he seems to have friends (even if just a few close ones) and be happy, generally.
For priorities, at the time our girls were growing up, I felt I was not successful in enforcing chores and citizenship in the home. DH was very conflict-avoidant and that didn't help, but I could have done more.
We did have our girls do their own laundry from an early age--boy, that was helpful since "picking up" could often involve tossing random clothes, even clean ones, in the hamper.
Also being engaged in regular volunteer activities should have the same weight, or more, than band or sports--something of choice, but "giving".
I'm saying that in the sense of, maybe you can't teach all of that, but still it seemed part of, life is teamwork and doing your share, and parents are not the kids' personal servants. And probably if parents just "wait" for kids to help out around the house, that doesn't happen often. And also, many kids could not do the 3, 4 or 5 extracurricular things if they also were helping with daily chores. Or, the family could brainstorm about, how can we work together to make this happen ? (Maybe it would require a major Tidying first!)
I actually don't think that's a bad thing for kids to see that figuring out how to do these activities and still eat healthy meals (many of my kid's parents tossed a MacDonalds meal at them on the way to gym or whatnot), keep the car running, etc and the parents also have some time for rejuvenating activities is a family concern.
So "pushing" to me is quite okay for those things, with significant consequences for not doing it, if example and encouragement aren't working.