You are right, and it works. Thanks Quinn!
Happy to say that notarytool runs on 10.15.7. For anyone else who is attempting this, I'll walk through my experience and hopefully that can help folks in the future.
First I'd recommend going to downloads here: https://developer.apple.com/download/all/ and downloading & installing the most recent version of Xcode that's possible to run on your OS (12.4 in my case). I think you need around 40gb of hard disk space.
To get notarytool, download Command Line Tools for Xcode (in my case, 13.4) From that file I'd use Pacifist (https://www.charlessoft.com/) to extract notarytools and install it. You could also try Quinn's method in Terminal, mentioned above. notarytool should be installed here: /Applications/Xcode.app/Contents/Developer/usr/bin/notarytool.
Open Terminal and send it to the directory that contains notarytool by entering the command cd /Applications/Xcode.app/Contents/Developer/usr/bin/
From here, you use notarytool by beginning your command with ./notarytool instead of xcrun (this last bit is why I was getting the error requires a newer version of macOS.)