You can do what you want with IMAP too.

I use isync, it's a CLI tool that can sync between IMAP accounts or between IMAP and local plain storage and it has the option to delete messages over a certain age after it copies them.

Is there a set up where some remote service handles email exchange, and you just sync with it

Any regular email provider can do this as long as it offers IMAP. I use email clients with my provider's IMAP but I also use isync on the server to periodically pull all email locally and delete messages older than 30 days. I've also installed a local private IMAP server and a private webmail client on top of that private IMAP, which I use to search through old emails.

This way I normally use the "live" mailbox with my PC/mobile devices, and use the private webmail when I need to search for anything over 30 days old.

What I do is not "selfhosted email", technically speaking, because I rely on the live IMAP and SMTP from a provider. But I can switch provider at any time because all my email is @ my custom domains, so I can simply point my DNS to another provider and update my apps to use the new provider's IMAP/SMTP server.