Are third party programs allowed to save screenshots unbeknown to the user in the first place??

In older Androids it was too easy for the app to get the permission to capture/record the screen without clear user consent. In newer Androids you need to ask permission at runtime.

So Chrome and Firefox should check if they're on a new or old Android.

It's only a part of the issue. The other part is that a "launcher" app shows miniatures of apps when you scroll through open apps, and this behavior is controlled by the same screen capturing permission. When you enable screenshots, you also enable previews in the launcher app when switching between apps, in some cases it might be not desirable.

There's probably some more nuances that I'm not aware of.

If the user gave it the permission too when they installed it yes.