Actually 20 people is around a reasonable number if you have some tech/expertise leads in the mix. 2 to 4 tech leads would be more than enough. Tech leads would be responsible for the technical decisions under their respective area and course setting for the future work but they don't do people management.

Managers do not need to mentor. Actually an independent mentor is better as there would be no conflict of interest. A person can get a mentor from another team as long as both are willing. If there are more mentees seeking mentorship than mentors, you can rotate the mentors or double/triple assign them especially to those who seek to switch to management track. In reality though, there are more mentors than mentees it seems. At least that's my anecdotal observation.

20 people management is very well possible. You don't need to have 1:1 with directs every single week. If you do it bi-weekly (every two weeks), the meeting cost to the manager would be 2 sessions a day. Managers tend to join every single meeting under the sun. That's how they fill up their time and appear(!) to be busy. In reality, they can cut down the meetings down to %25 and nothing would change.

Let's look at lowest level managers. They can meet 20 directs bi-weekly for 0.5hr. Let's add 0.5hr follow up work after each 1:1, which is way generous. So that totals up to 20 x (0.5 + 0.5) / 2 = 10hr. Let's assume their own 1:1 with their own manager to take 1hr bi-weekly or 0.5hr weekly as they need to discuss both team and personal issues. They can meet with peers and their manager weekly for 2hr. They can meet with sub-teams (let's assume max 4 teams) weekly (4 x 1hr) for status and problems. And at the end of the scrum (bi-weekly), they can meet all the directs for demos and postmortem for 2hr. If there are some other technical meetings they do not need to attend unless they are absolutely needed but this shouldn't be often. Leads are perfectly capable of handling those and report back if they're hesitant, which, again, shouldn't happen often. Meeting tally up should be around 18hr. For a 40hr week that's less than half. Let's assume they're dragged into some technical meetings for 5 more hours that's still 23hr. Remaining time should be more than enough to sifting through emails, quarterly duties (e.g promos, end of quarter status, next quarter strategy/goal setting) and occasional emergencies.

For mid-level managers (generally directors), they don't have to meet with teams but we assumed their 1:1 would cost 0.5hr/week. Their follow-up should be much less though as they should be able to make decision on most of the issues on the spot. But let's add a quite generous %50 on top which would sum up to 0.5hr x 20 x 1.5 = 15hr. I think they should meet with tech leads (skip-level) too if they're one removed from the lowest level. They can meet them for 0.5hr every 8 weeks which should take at the very maximum, 20 x 4 (leads) / 8 * 0.5hr = 5hr/wk but it'll probably take less. Tally up is 24hr (if you add peer & direct meetings). Mid-level managers may get more randomized but they need to be efficient at what to get involved and what not to. Majority of the middle-managers make the mistake here. They get involved more than they can chew as they're micromanagers and/or they respond to every single demand from their own manager, practically doing the manager's job for them. This is because their promo is generally tied to their manager's attitude toward them but this should be changed. How you can change this is a separate topic I can get into, if needed but let's skip it for now.

For high level managers (generally VPs), their time would be spent less on skip-1:1s & technical meetings but more on strategy and status meetings. I believe they would have even more time at their hand than directors if they're vigilant on the time sunk on the emails.

Then you have CEO at the top or you can have SVPs if number of your ICs is above 16K. But, yes 20 people management is possible if managers are cognizant of their time and boundaries of their responsibilities and if they actually work instead of appear to be working.

But let's face it, it ain't happening because managers are
A) either people who doesn't want to work anymore but need the money so they'll just carry on until they f-up big time and then switch companies and rinse & repeat,
B) or, they overstep their boundaries,
C) or, they don't know how to do prioritization and time management,
D) or a combination of some/all above.