Sorry.
Seems like I just learned something new.
I have not noticed up to now, that MyPhoneExplorer does not backup my SMS.
Thx a lot for that info; I never would have guessed that.
Edit:
Just remembered this thread from last year:
Recommended by some users were QKSMS and Signal.
I just have no idea, how they do handle the database.
See also this thread as a continuation of the first one: