It works like this:
1) Install AHK as normal
2) Launch the app, click "Backup Current Version".
This copies the AHK exes from the install folder to C:\Program Files\AutoHotkey\AHK-EXE-Swapper\Versions
3a) If a new AHK_L version comes out, download it and run the installer.
Then go into the app and click "Backup Current Version" again.
3b) For AHK_H releases, or AHK_L betas (Without installer) and maybe this applies to v2 AHK as well?
Copy the zip into C:\Program Files\AutoHotkey\AHK-EXE-Swapper\Import
Then open the app, click "Refresh all", then select the zip, then click "Import Selected Version"
4) Once versions are in your library, you can select them from the top list and select "Replace with selected version".
So basically, the mistake you are making is that it does not rely on files in the C:\Program Files\AutoHotkey folder, but it stores "Other versions" in C:\Program Files\AutoHotkey\AHK-EXE-Swapper\Versions
You populate versions by overwriting ALL the current EXEs (AutoHotkeyA32.exe, AutoHotkeyU32.exe, AutoHotkeyU64.exe) with new ones, and then clicking "Backup Current Version".
This will add it to the "library" of versions.
But if I try to run AHK-EXE-Swapper as administrator, it doesn't work for me.
You do not need to do this, it already self-elevates to admin.