How is it possible for anyone to manage 25 people? I see maximum 5-7 reports before it becomes overloading.

You delegate all responsibilities other than information collection and administration. (That is, approving vacation and the like.) You trust the most senior people below you to tell you when there's a problem. You rely on gut instinct over diving deep. And you accept that your organization's performance is degraded due to the extra overhead.

I did it for a few months. It was truly unpleasant.

Yes. I have seen successful teams with one manager having 20-30 reports.

It does not happen by accident. Both the manager and the reports are usually top notch for the team to be successful. The manager knows how to effectively delegate responsibility (not tasks) and the reports are intelligent, self-driven, reliable and know how to use their network for help and report back to manager when they are stuck.

Best test if the manager is able to have more reports is if he/she can go for 2 week vacation and the team is still delivering. If the manager hoards operational responsibilities then this is probably not going to work well and he/she is not ready to have more reports. To have 20-30 reports you need to delegate all operational responsibilities so that you are not in critical path of any ongoing project.

It takes a lot to learn how to be accountable without being responsible. Further, you have a lot of work to invest in a team that may not have the skills necessary - meaning that you accept reduced quality or put in extra hours of mentorship over just doing it yourself. Further, if your team is doing delegated work, they aren't doing IC work, and it's hard to justify at times.

Over 15 reports, though, you have no choice.

You have choice. You can do shitty work. Which is what I see in most cases -- managers happy to take on new reports but unable to actually manage them. Work piles up and progress grinds to a halt as the manager becomes a severe bottleneck.

Par for the course is blaming that their team is too large and the team is broken up and they become a director with 2-4 reporting line managers.

I don't think it's referring only to direct reports but to the entire reporting tree under you.