@jlfranklin mentioned Database SSL Check in https://github.com/backdrop/backdrop-issues/issues/4772#issuecomment-778.... That is a D7/D8 module, and it could of course be ported to Backdrop contrib, but it's only a ~40-line implementation of hook_requirements(): https://git.drupalcode.org/project/database_ssl_check/-/blob/7.x-1.x/dat... ...so I believe that it could be a nice addition to core, as part of #3624.
I've installed the module in Backdrop (simply modifying it's .info file), and this is what it adds:

Recent comments
Thanks, the /tmp folder was the problem. I moved it elsewhere and then the updated worked. Thanks again!
Update 1.35.1 appears to install but hasn't
Probably not the reason, but worth to do a quick check: Sometimes the update cache interferes, so if something's odd with updates - some module (or core) claims to have updated, but then...
Update 1.35.1 appears to install but hasn't
Nope, not touched any permissions!
Update 1.35.1 appears to install but hasn't