I used Nextcloud but now, I just forward copies to an extra email box, since I can reply to texts from there, which you can't do with Nextcloud, called SMS Gate. Get it from F-Droid.
F-Droid
F-Droid is an installable catalogue of FOSS (Free and Open Source Software) applications for the Android platform. The client makes it easy to browse, install, and keep track of updates on your device.
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Nextcloud yes, Google, might as well ask for a direct data link the the NSA data warehouse in Utah.
I like the SMS gate app from f-droid suggested.
A different approach to read SMS from other devices, for example your PC, would be using kdeconnect. It's supposed to allow you to read current messages, receive messages, and reply to messages, even send messages I believe. So perhaps an option, though not a syncing tool to keep SMS messages somewhere else, and I can't tell about the encryption involved...
You can try and go Asteriks way, get a 3g modem, and be able to call and send sms
Would join, by joao apps meet your needs? It should get the job done, but I'm not sure how much you would trust it.
From the description on the Play Store: Yes.