There have been many issues reported with Mission Control on macOS 27 Golden Gate, especially regarding the way Full Screen apps are handled. Some have been resolved by identifying and removing incompatible 3rd party extensions, others are bona fide bugs.
There have been many issues reported with Mission Control and Spaces on macOS 27 Golden Gate, especially regarding the way Full Screen apps and app switching are handled. Some have been resolved by identifying and removing incompatible 3rd party extensions, others are bona fide bugs.
Start up your Mac in safe mode - Apple Support
If that does not correct the issue, test in a new user account. A newly-created user account will have all settings at default, and if the issue does not occur in that new account then it provides guidance on what may need to be changed to correct the issue.
Add a user or group on Mac - Apple Support
Once the testing is complete, the new account can easily be deleted.
Delete a user or group on Mac – Apple Support
Those steps will help to isolate the problem.
It might help to download and run EtreCheck, written by a long-time forum member. The free version is fine. Post the full report using the Additional Text button. No personal information is revealed.

If the issue persists in safe mode and a new user account, that suggests that it's a bug that Apple needs to address. Bugs can be reported to Apple here:
Feedback - macOS - Apple
Personally, I CAN replicate the issue you see on my Mac running Golden Gate, where with app grouping on in Spaces, trying to drag a group by the app icon just drags the top window and not the whole group. It works as expected on my other Mac running Tahoe. I posted screenshots from Tahoe and Golden Gate in a prior thread:
Mission Control Mac OS 27 - Apple Community
Reporting issues like this to Apple is the only way to get them fixed. They do prioritize bugs based on frequency of reports, so the more people who report it, the faster it will be addressed.