Instead he would spend his day pairing with different teammates. With less experienced developers he would patiently let them drive whilst nudging them towards a solution.
That's not a management activity - that's the kind of coaching you would expect from a senior IC ("Individual Contributor" - I still hate that term.)
Generally I would expect a "manager" to have authority over other people in the company: run performance reviews, handle promotions and hiring and firing.
I think it's important for companies to provide a career structure that allows for influence, decision making and leadership roles that don't also require taking on those management tasks. Management tasks are extremely time consuming and require a substantially different set of skills from being a great team coach or force multiplier like Tim in the story.