I sync my emails using a Dovecot IMAP server on my home server. I fetch emails from all my accounts with fetchmail and sort them into the right folders using Sieve. They get indexed and are searchable (ultra fast!).
225000 emails in 13 GB (ZFS; uncompressed 18 GB).
Basically what I said. Dovecot can be installed on every Linux or BSD system. You’ll need Pideonhole and FTS Flatcurve as extensions.
When you install fetchmail, you can let it connect to all your IMAP or POP3 servers. Each process will deliver your mail instantly to your own Dovecot server.
You’ll also need a certificate for Dovecot. This can be solved using Letsencrypt.
You can use any mail client you want. I use Fairemail on Android. On desktop and notebooks it’s Thunderbird.