I'm a first-time Director at a SW company with a total org under me of ~30 people in 5 different subject areas. I struggle with what you highlight, but it's impossible for me to go deep in all these areas. MY boss is the CTO and he talks about "T-shaped" or broadly across and narrowly deep. I really don't like this, but the reality is I view myself as senior-dev level in one area, int in another, junior in 2, and barely familiar with the third - and I'm by far the most technical of all the Director/VPs