I am using my own Mail Server (mailcow) beside my old gmail and other adresses. But for convenient, searchable offline Backups, I use Mailsteward (macOS) ... if it is of interest for someone: https://mailsteward.com

(alternatives for other OS: https://alternativeto.net/software/mailsteward/)

Then I put the database on multiple backup locations regularly.

Another thing, some people do not already know: If you don't need a throwaway-adress for some services, and you just want to make your mailbox more structured, you can use '+' before the '@' to add another word to your email adress.

Like: your.name+randomName@gmail.com

The +randomName will be ignored and the emails are received at your.name@gmail.com. But most Servers (I use) will put a '[randomName]' before the subject of the received email. Which can be quite handy for handling your emails. Even more, if the company uses multiple different adresses to send you emails.