According to Adobe, Elements 2021 should work with Monterey if you have updated the OS to version 12.1 or above. If you simply upgraded your OS to Monterey without reinstalling the program, I suggest you uninstall and reinstall PSE 2021. (I know virtually nothing about Macs, so don't know if this is how an OS update works.) If you are still having problems launching the program, try contacting Elements support. You can start a chat session by clicking on the blue Contact Us button under the Support menu at Adobe.com. Or you can contact Support by telephone using the numbers listed here: https://helpx.adobe.com/contact/phone.html.
Of course, I don't want to dissuade you from updating Elements to the 2022 version if you choose to do so.