Crymaracmur Said:
"That's not it either... It's not that I can't see my purchases in that tab, it's that it can't open a connection on that tab at all.: I spoke with Apple tech support on the phone today and they seem to think this is just an OS issue. At a certain point between High Sierra and Big Sur, they say that the Purchases tab on the App Store app on the Mac stopped working. The only solution is to upgrade to Mojave, then if it still doesn't work, go up to the next one and so on."
-------
32-Bit-Only being the Culprit:
Catalina and thereafter is 64-bit only. So, in Big Sur, if these are 32-bit applications, then they just will not open. So, contact the developer, requesting a new install package for 64-bit version. Also, perform a reinstall.
In Depth:
Three Things:
A. Big Sur is 64-Bit Only:
The key issue here seems to be compatibility. What did you upgrade from? Was it pre-Catalina(macOS10.14 and before)? If so, note that 32-bit apps are not supported on Catalina and Big Sur. So, you may have 32-bit apps installed.
B. Contact the Developer:
If you don't want to go though the hassle of downgrading, then contact the developer, seeing if they can give you a new license, and if they have 64-bit installation versions to use.
Also...
C. Reinstalling the macOS:
Another option would be to reinstall the macOS. Perhaps the install was not 100% successful? Sometimes, when major changes are made to your Mac(i.e. update installations), system settings (i.e. for Mail) become manipulated, technically. So, when reinstalling, nothing would be deleted. It's just that items that have become corrupted or deleted, would be replaced or reinstalled back to where they should be. Go Here: How to Reinstall macOS from macOS Recovery - Apple Support
Important: First, Back Up your Mac:
Create a Time Machine backup of your Mac, so that you can have something to restore your Mac from, should anything go wrong with the reinstall. Take this as a learning experience, knowing to always create a backup of your Mac, prior to making any major changes to it (i.e. update installation)