Apple would not warn you about updates provided by 3rd party developers, they have no way of knowing if the update provided for a 3rd party app is compatible with you system or not. Though in general it should not have been able offer an update from the App Store that does not work on your operating system as that is specifically defined for the update and the App store would not offer it then.
The other option, is the update is compatible with your computer, but its just glitched out or something else happened. You mentioned "it ceased to function". But did not explain what that means exactly. What happened when it updated, and whether or not you have tried to force close the app, and attempted to open it again.
Whether you can update your Mac depends on what model / year of Mac you have. macOS 11 Big Sur is compatible with the computers listed in the following link.
click here ➜ macOS Big Sur is compatible with these computers - Apple Support
You can get a copy of macOS 11 from the Mac App Store app, by using the link in the support document linked below.
click here ➜ How to download and install macOS - Apple Support