Like you said, banking apps. The logic behind that is they use google to security check their apps. A random non-bank example would be the slick deals app. Without play services it would just open then crash.

Many apps use play services for their notification system. So for instance, proton mail works fine but notifications do not.

NFC is not supported, so anything that uses that won't work.

Not an app, but I was surprised that widgets don't work unless you're in the primary profile. Technically they work on any profile, but they randomly get deleted, and frequently. It's a known bug that probably will never get fixed because the source of it comes from stock android.

I will mention that you can have a profile running play services, which gives you access to many apps that wouldnt normally work. And it's sandboxed so it has less impact on your information (I don't know all the specifics but it does limit in some way how much it can snoop into the rest of the OS). Then you can also set up granular controls on your apps to limit them from snooping.