In KMail I have the "download messages for offline use" option, accessing cached messages offline isn't an issue.
Remove from server is still an issue, but again "archive" plugins exist. (that is NOT user friendly, I'll grant you that).
Thunderbird has both these options too. AFAIK the "full cache" is called "synchronization"
I use IMAP because I have my main machine with offline cache and all the archiving bells and whistles, and then I want access on my phone, with synchronized /READ flags. POP can't do that.
IMAP also supports server side search, so even mobile clients can search the whole email database quite well. (except encrypted mail, few and far between)
I agree that IMAP's most dangerous feature is the server propagated message deletion. But if you don't trust your provider to not do that, you have bigger problems to solve.
Use quality mail clients; on mobile that's a challenge...