Hello SgtSurehand an all the others here in the thread,
I've never used google servives at all and I don't need my device to show emails immediately. So I really don't have experiences with these settings.
But, just for my understanding and maybe as a proposal for you, why don't you use for example any emailprovider together with FairEmail (K9-Mail) and allow FairEmail to notice you when new emails arrive?
I have just tested it: I changed the settings in FairEmail and allowed it noticing me when new emails arrive. I sent an email from my desktop computer to my email account on my pixel and ~ three seconds later my pixel beeped. Is this different to push notifications? Where are the advantages?