"There's a very specific reputation I want to have on a team: "Nat helps me solve my problems. Nat get things I care about done." That's the reputation that's going to get me the results I want in next year's performance review. That's the reputation that's going to get me a referral a few years from now."

This works for a referral, but it is a career killer. I did this for 20 years in different departments. Every time they wanted to keep me there to solve their problems, but this never puts you on the promotion list. I had a colleague that retired after 40 years, he was the go-to person for any problem, but he was never promoted, he was way more competent than any of his managers and their peers. He was the ace in everyone's sleeve, rarely recognized (it was considered to be "normal" that he solves any problem) and never rewarded; his performance was considered an expectation, while the rest of the people at the same level had a much lower bar. In the last performance review, his manager said that his performance is compared with his goals and targets, not with peers. I was told the same by a different manager, so it is not an accident.