it’s possible to have an email client download all the messages from Gmail and remove them from the server. I would like to set up a service on my servers to do that and then act as mail server for my clients. Gmail would still be the outgoing relay and the always-on remote mailbox, but emails would eventually be stored locally where I have plenty of space.

Do you really need this extra server? Why not just configure the account on Thunderbird and move the older / archival mail to a local folder? Or even drag and drop it out of Thunderbird to a folder and store the resulting files somewhere?

I’m just asking this because most people won’t need regular access to very old email and just storing the files on a NAS or something makes it easier.