Is there a set up where some remote service handles email exchange, and you just sync with it eg through POP style interfaces?

Yup, that’s signing up for a email provider. I wouldn’t call that self hosting since you’re just referring to email client software on your end to store the emails that are being synchronized at the email provider’s server(s).

IMAP is sync so that might be what you meant. POP is even older but that’s for downloading only. And SMTP is for sending.

Those all still exist, most paid email providers still offer them just make sure to check before signing up.