They are different jobs, but I think still being an engineering manager benefits from having spent time as an engineer, more than the other way around.

I have to somewhat disagree. Being an engineer makes you a better engineering manager. However, that puts a floor on how bad you could be. It doesn't put a ceiling on how good you could be. I've had great managers who never were engineers. However, the floor for managers who have never been engineers is lower than the floor for engineers who have been.