Depends on the service. For example, if proton bans me I can move my email to a different provider because I own my domain. In case of Obsidian, it's an offline app that uses Markdown, so I can easily migrate to a different note-taking software for any reason.
Your backup note is on point; I highly recommend everyone to do a Google Takeout every few months (or at least years!!). If you've never done one, like most Google users, you're playing with fire.
Notice that Google Takeout is broken with regards to non-UTF8 email encodings (for some of you who archive old emails). I noticed it when coding a small tool https://github.com/karteum/gmvaultdb to (among other things) import the mbox from Takeout into an sqlite DB : Google performs some encoding conversions that permanently break all non-ascii characters in non-UTF8 emails (they are all replaced by 0xEFBFBD). In order to archive old emails, I had to use mbsync with IMAP (gmvault does not work anymore with Python 3, and I didn't try yet https://github.com/GAM-team/got-your-back ).
I think for email it makes sense to have an offline IMAP setup (i.e. a copy of all mail is downloaded) on at least one device (e.g. your desktop) so that you can still access things if you are abruptly cut off from the provider. This can come in handy even for temporary outages.
I agree, but an an alternative to mbox/maildir format on your machine : I wrote https://github.com/karteum/gmvaultdb in order to have a single-file SQLite DB with all old email contents (all UTF-8 inside, simple/minimalistic schema), together with a small GUI to consult emails and do queries (full-text not fully implemented yet but this is the target) and all attachments extracted in a side folder so that I can directly browse with a file manager.
> If your domain registar bans you, you’ll lose your email all the same.
Maybe? I would hope that you could raise the issue with the dowmain registry or ultimately ICANN if the registrar does not let you move the domain elsewhere after banning you. Registrars do not have the same platform ownership over domains that Google has over gmail accounts, they are just middlemen.
Your backup note is on point; I highly recommend everyone to do a Google Takeout every few months (or at least years!!). If you've never done one, like most Google users, you're playing with fire.