Also I have experimented a little and found something more. If I delete Google Account from the phone, force stop Hangouts, clear its cache and try to log in with that account again, Hangouts shows this screen [1] with "Back up to Google Drive" option enabled again.

I tried twice - and every time Google sets this option on by default even if I had disabled it previous time.

Of course, this might be not intentional - maybe the enable sync flag is not stored at Google server.

[1] https://i.imgur.com/7XjfCst.png