This started out as me trying to ask for help, but in the meantime I figured out what was the problem, so I'll post this as a mini guide for those who might run into the same issue. I was trying to install Flash Player on macOS, but it didn't get recognized among the plugins. Then I noticed that the
Flash Player Download Center kept giving me the PPAPI (Pepper) plugin. Darn!

The direct download link for the latest NPAPI plugin is here:
https://fpdownload.macromedia.com/pub/f ... er_osx.dmg
In case you already installed the PPAPI plugin by accident, you can get the uninstaller here to scrub it first:
https://helpx.adobe.com/hu/flash-player ... ac-os.html