Whenever I downloaded a new .dmg (in this example I will call it myprogram.dmg) file and tried to run it nothing happened. I tried rebooting, updating (to 12.4) and control+click -> choose DiskImageMounter but it refused to open.

To get around this I opened Terminal (either search for it or open via Finder -> Applications -> Utilities).

Change directory to where you downloaded dmg file. In my case It is in my download folder

cd /Users/jordansphere/Downloads 

The run the following command to mount the dmg file for installation

hdiutil attach myprogram.dmg

You can then install from there.