What I do is use Claws Mail with POP3, it has an option that allows a message to only be deleted from the server after a configurable period of time. So if you set it for 10 days for example the message will exist both locally on your PC and on the server for 10 days, after which it will only exist on the PC.
It works pretty well in general. The only account giving me some trouble is Yahoo, which I suspect has some quirks, which occasionally cause the messages to be downloaded again and duplicated. Thankfully it’s easily fixed because Claws also has a feature to delete duplicates.
This approach is different from IMAP, which would maintain a local offline cache of the live inbox, but you wouldn’t be able to only keep local messages — any change in one side would be reflected in both.
However, Claws allows you to do both. You can have both a POP3 and an IMAP account connected to the same live box use the POP3 for offline archival, and the IMAP for when you want to put something back on the server, or if you need to look at other folders on the server besides inbox (POP3 cab only see the inbox, not trash, sent etc.)
Normally I only do folders locally on the PC, on the mailbox connected with POP3, so none of the organization is reflected on the live mailbox, which is inbox only. Every once in a while I connect via IMAP to recover emails from the sent folder, which I’ve sent with webmail or from mobile (using IMAP on mobile too).
If this doesn’t fit your workflow turn there are lots of IMAP syncing tools like you’ve noticed. IMAPsync is pretty good.
The last step for my workflow would be to self host an IMAP server that will index the POP3 mailbox, and expose it read-only (without SMTP) through a webmail app, for archival and search only. I may have to look at Piler. The quirk here is that the Claws mailbox format is slightly different from IMAP, it’s very similar to mbox but not identical, will have to see if any IMAP server will accept it.
Thunderbird is no go unfortunately, its main box format is to keep all messages on one big file instead of individual files, which complicates things a lot.