I believe you can use the command-line tool "adb" for this. It's been a while since I've done it since I use syncthing now, but here's a quick intro:
- on phone, enable developer options
- on computer, install android tools, which should come with adb (Do not install full android developer toolkit, it's huge. You only need adb)
- on computer run 'adb devices', which should currently print that no devices were found
- plug phone into computer, and accept authorization prompt on phone
- on computer run 'adb devices' again and it should list your phone with the status "authorized"
Now you should be able to use adb pull -p -a <folder name> to back up folders to your computer, for example adb pull -p -a /sdcard/DCIM.
Note that you will have trouble backing up the /sdcard/Android folder, which contains a lot of app data. Unfortunately I think you need root to back up that folder. Alternatively you can back up individual apps using adb backup ....
Personally I think in the long term, automatic backups like syncthing is better. Manual backups are a hassle and because of that, you won't do them as often.