I've worked as a TL on several teams before where we didn't assign any implementation work to me specifically, but I was there to jump in and help paddle the boat when folks got stuck.
This worked great, because usually the sticky places were integration points between subsystems, which can be especially tough for junior engineers to navigate. As someone with the big picture I could jump in and save them tons of time.
In the meantime, if I had spare cycles I'd work on tooling, test automation (and especially stubbing out and deflaking integration tests).
Projects where I did this tended to have fewer bugs and hit their deadlines.
What I referring to is a team of 50 people, say, with 10 of them that are "on rotation" sitting on the bench and wait for new assignments. What happens is that 2 of them get used to the "vacation" and are not much motivated to get on a new project when the time comes, 2 disappear, 1 takes advantage of the R&R to work on tools and do some self-study, and 6 start interviewing for other companies. Horses want to run.
As for (2), that's as true as fairy tales.