Ended up solving this myself but thought it would be a good idea to share my solution in case anyone stumbles upon this thread with the same question some day. My solution was to first make sure all my accounts were signed in (in my case my Apple ID and my GitHub account, not sure if the GitHub account made any difference tho) since it seemed to be a signing issue. Then I cleaned the build folder (Product -> Clean Build Folder or Command + Shift + K), and deleted the Data folder at /Users/campbell/Library/Containers/com.NCX-Programming.SwiftUI-Test/Data/. Next I went into the "Signing & Capabilities" settings screen in my project settings, disabled "Automatically manage signing", turned it back on, and allowed Xcode to reset all of the signing settings. After that I just reselected my team and previews started working again.

Ok turns out my issue was caused by failing RAM, replacing it solved my issue (I have a MacBook Pro that has user-upgradable RAM). Probably not related to the issues in Monterey.

Having this issue too, though without the popup and on Big Sur. I've discovered that a sudo purge works as a quick fix, though it ends up coming back. You can run it over SSH too if you can't open Terminal anymore. Hope there's a real solution for this soon.