I remember interviewing someone once that claimed to have 200 direct reports. I was really blunt and said "nobody has 200 direct reports".
They assured me that they did. So I grilled them on the mechanics of it.
If you had biweekly 1x1s with all of your DRs, you would have to do 100 per week. If 1-on-1s were your fulltime job you could at most do 80 per week (2 per hour x 40 hrs). Thats just rotating through 1x1s. That's not doing anything else.
He claimed "team leads" did 1x1s, which is fine, but its hard to call them a direct report if you're not even aware of their daily work. Theres just no way you can adequately manage 200 people. As mentioned above, you are just a glorified HR person, you're no longer in engineering.
I would never hire a manager below Director level that isn't themselves an individual contributor. I know its possible to have that but I wouldnt hire it, because we expect IC on our teams.