I set up some pop3 a/c's in Gmail and it does exactly what I want. Thanks.
However, there is a bug, or is it simply misrepresentation, in Gmail relating to pop3 a/c's. For each pop3 a/c set up there is a Settings page where you can set the Sync period, including Never Sync. You might conclude that this would turn off sync for that a/c. It does not. Instead you have to go into Google Settings/Accounts and turn off sync for the pop3 a/c. Only then does it not sync. This is such a fundamental bug that I'm surprised it has not been identified and fixed before now (Android 5.1).
I think this is backwards in terms of what you originally said you wanted to do, assuming I understood: stop emails from sync'ing to your phone. I've never tried this because--I like to sync. I need my mail. But your concern as I understood was not a question of what syncs TO Gmail, but how GMAIL syncs to your Android device.
The sync commands for individual POP3 accounts only control when mail shows up in
all aspects of Gmail, including your desktop browser. That's a completely different issue than when you sync gmail to your Android device. You can--again, I believe--allow gmail to sync normally with your pop3 accounts (and should...so your mail is available promptly when you are home and on your desktop). Then, everything is in Gmail when you want it.
Now, you're now no longer dealing with POP3 issues. You're just dealing with what it is in gmail (everything collected, pop3 or not) and when that goes to your phone. That's a different sync issue. Go to GMAIL on the phone, use the instructions in my first post, above. (#2 in the thread). No matter how the the POP3 account syncs to
Gmail, those messages and others in gmail won't download to your phone if you don't allow gmail overall to sync to your phone.