I attempted to use SMS Gate (a fork) to do backups from the stock messaging app to the SD card. It didn't work.
> I did hear of some folks using their own IMAP server ...
I ended up syncing to a Dovecot IMAP server on my local Linux desktop, which seemed to work just fine. I was apprehensive about the complexity of setting an IMAP server up, but that turned out to be misguided - Dovecot was incredibly simple to get up and running. I assume SMSBackup+ would work just as well as SMS Gate did here.
I eventually ended up moving from the stock messaging app to QKSMS (GPLv3, Github, Google Play, F-Droid) because it has built in backup functionality.
How about an app that just, well, sends the SMS texts to your email, as regular emails? A filtering rule would put them under a certain tag and skip inbox.
The upside of the design is that it requires only one, very clear, permission: send emails to a given address.
SMSBackup+ makes the sender the contact / phone number and the receiver as your number. It's been fantastic for archiving all of my texts and such. I've been able to search it all relatively easily.
I reached out to the author because I've been a user of the app for several years, but I haven't heard back (this was only a few days ago). I'm hoping it can continue in some form.
I'm very much hoping that we could resolve the issue with SMSBackup+ for the time being, but that's mostly up to the project owner.
I did hear of some folks using their own IMAP server and CALDAV service to target instead of Google, but I have not tested it.