This is the correct answer. The manager's job is to manage the people, not the project. That's what a tech leads and PMs are for.
A good people manager works to keep their employees comfortable, aims to avoid conflicts (and resolves them if they arise despite these efforts), mentors and coaches people to develop new skills, and shields them from politics. Doing the work of an individual contributor just gets in the way of that. The manager should be familiar with and understand what their people do, but leave the actual performing of that work up to them.
A good people manager works to keep their employees comfortable, aims to avoid conflicts (and resolves them if they arise despite these efforts), mentors and coaches people to develop new skills, and shields them from politics. Doing the work of an individual contributor just gets in the way of that. The manager should be familiar with and understand what their people do, but leave the actual performing of that work up to them.