Taking on big stuff isn't necessarily a "big ego" thing, but getting upset that we didn't solve the problem might be.
In my experience, big problems get solved by teams. It's important to be a good team member, even if we are "close to the gods." From what I understand, Feynman seemed to work well in teams, but I never worked with him, so I don't know.
False humility isn't helpful either.
I have worked with, and known, many truly brilliant folks. Some were unbearable, and getting on their team meant you drew the short straw.
Others were so self-effacing that it was actually embarrassing. It was difficult to be on a team with them, as we often accidentally treated them disrespectfully, or ignored their input.
I think it was Mandela that said: "You are a child of God. Your playing small doesn't serve the world. There's nothing enlightened about shrinking so that other people won't feel insecure around you."