(https://hbr.org/2013/12/how-google-sold-its-engineers-on-man...)
I am an Engineering Manager/Director and my life is quite hectic and not because of busywork. You work with some very brilliant people, but they often fail in conflict resolution. Even if you don't have EMs, you still have upper management and more often then not they're out of touch, here the EM acts as a translation layer that understands both the business and the tech enough to balance these tensions well to make sure the team works well while delivering value for the company. Sometimes you need to deal with other teams with their own agendas.
In essence - if things go well, there's less need for an EM. I had a well running team last year, so much so that my life was quite calm. Then the leadership forced product domain and technology change on us and we lost our tech lead. Even if 90 % of the team stayed the same the dynamics were completely new and I had to help the team navigate an uncertain environment. Things went to normal and suddenly we became responsible for the core of a cross-team project and I am busy pushing against leadership, getting other teams to commit to take the burden off of our core team and other much needed activites. My tech lead couldn't do that while also designing the tech solution so in a way I am helping them do better work.