Hazel 5.1 "Check for Updates" doesn't detect Hazel 5.1.2

I'm running Hazel 5.1 on Monterey 12.4.
When I noticed in MacUpdater that Hazel 5.1.2 is the latest release, I decided to let Hazel update itself. However, neither "Check for Updates..." in the Hazel application menu nor "Check Now" in Preferences > Advanced offers to update to the latest release. The only thing that happens when I do a Check is that progress bar appears for just a moment, and then both Check functions are disabled until I quit and relaunch Hazel.
The one thing that's a bit unusual about Hazel on my Mac is I've installed it in a "3rd party apps" folder instead of the macOS Applications or Utilities folder.
I realize I can download and install the latest release. But I'd really like to have Hazel's Daily Check for updates to automatically detect a new release.
-- Ward
When I noticed in MacUpdater that Hazel 5.1.2 is the latest release, I decided to let Hazel update itself. However, neither "Check for Updates..." in the Hazel application menu nor "Check Now" in Preferences > Advanced offers to update to the latest release. The only thing that happens when I do a Check is that progress bar appears for just a moment, and then both Check functions are disabled until I quit and relaunch Hazel.
The one thing that's a bit unusual about Hazel on my Mac is I've installed it in a "3rd party apps" folder instead of the macOS Applications or Utilities folder.
I realize I can download and install the latest release. But I'd really like to have Hazel's Daily Check for updates to automatically detect a new release.
-- Ward