My question title ended up being misleading. I meant "without a Google account" i.e. Play Store. It wasn't that I didn't want to use an app, and that's actually what I ended up doing in the end.
In case it helps anyone else, here's what I did. After a lot of research, I found there's a free app called "SMS Backup & Restore". You can download the APK (my phone is old enough that the browser doesn't work on most websites due to not having TLS, so I downloaded it onto my PC and transferred it via cable. This method works just as well!) This is the one I used for Android 4.1.2: https://www.apkmirror.com/apk/synct...s-backup-restore-9-75-1-android-apk-download/ but there are also newer versions, of course.
I just ran this file with my Package Manager, it installed the app, and I was able to backup all of my text messages very easily. They save in an XML file, which I've now copied to my PC. Gonna write a little program so I can browse them on my computer 
I wasted a lot of time downloading desktop programs that were supposed to be able to back up text messages and were advertised as free downloads, but once you download and install the program, you find out it's just trialware and it will actually cost you upwards of $70 USD/year to use! Don't bother with those unless your phone is otherwise unusable, SMS Backup & Restore works great
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