Wouldn't Spider-Man's ideology land him on Cap's side?
When Tony Stark recruits Peter Parker in this movie, Peter says "When you can do the things that I can, but you don't, and then the bad things happen, they happen because of you." Doesn't that explain majority of Cap's argument? He says "What if there's somewhere we need to go and they don't let us," as well as "If I see a situation pointed south, I can't ignore it. Sometimes I wish I could." Peter and him share the same type of burden if they don't help people that need it.
I really feel Tony left out quite a bit of information when recruiting Peter.