Edit: I had a much longer post than this which had edits as I learned more and corrected my previous conclusions. I've now realised that the problem is really that the High Sierra update has wiped every password from my keychain. It's not a Mail problem. Keychain has no remains of any of my passwords. iCloud asked me to sign in when I start up my laptop. Mail says it can't access any email accounts and asks me for my passwords. Websites in Safari with saved login details require me to login again. It may have been all of keychain that was wiped, not just the passwords. Either way, this is a nuisance and feels really sloppy of Apple to allow the possibility of this happening.