Your story does not define which management roles you are writing about.
Program Management
Department Management
Project Management
Director
VP
President
CEO
CIO
CFO
It seems you might be talking about Department Manager or Line Manager
Sales Manager, Design Manager, HR Manager, Production Manager.
First tier manager, direct reports don't have direct reports. The bank with 12,000 employees and only three levels of sounds like fiction. Top Man, middle managers and workers. I approve of a flat management structure but that is too flat, not good. A CEO with 100 direct reports who each have 120 direct reports does not work. Now if you add a level your can have a CEO with 25 direct reports who have 25 direct reports who have 20 direct reports that works, I consider that almost ideal. That might be what was meant the three levels (either not counting the CEO or not counting the workers as a level.) In that structure out of your 12,000 employee's 626 are managers or just over five percent of the workforce. You have to be a really big company to achieve this as smaller companies individual departments often are run by a working manager with a tiny staff they might have a flatter organization, a real three level organization but most line level managers have less than 20 direct reports also do line work and report directly to the CEO so when you count everyone who has direct reports as management then a company with 100 employees might have 20 managers or 20 percent of the workforce. Most businesses are small businesses with very flat management structures but a high percentage of their workforce has management responsibilities but also line work responsibilities. I have seen Restaurant Managers busing tables, bringing orders to tables, seating customers, filling in where ever needed.
Seems like almost every work/business related story here presupposes everyone works in a cube farm/silicon valley open plan office if they are not working from home.
Note to the world, most workers do not have a desk, they work on a factory or retail space floor or outdoors/underground. If they are sitting down it is in a piece of machinery like a tractor, delivery truck, forklift, backhoe, bucket truck etc.
TEK