Your description identifies a specific change: since updating to macOS 27, files displayed from a Desktop Stack do not consistently drag into an open application such as Safari.
Apple's macOS 27 documentation confirms that Desktop Stacks can be expanded to display their individual files.
It does not indicate that macOS 27 intentionally changed the ability to drag those files to another application.
A useful way to narrow this down is to test the same screenshot using different methods.
First, wait until the screenshot has finished saving and is fully visible in the Screenshots Stack.
When the problem occurs, try dragging it from the expanded Stack to another location on the Desktop.
Then temporarily turn off Stacks with View > Use Stacks and try dragging the screenshot directly from the Desktop into Safari.
You can also open the Desktop in a Finder window and drag the same file into Safari.
If the problem persists, restart the Mac and repeat the tests.
If dragging from a Finder window works consistently while dragging from the expanded Stack does not, that would provide useful information about where the problem may be occurring.
I would leave more extensive troubleshooting, such as deleting Finder preference files, until these basic tests have been completed.
At this point, there is not enough evidence to identify the behaviour as a confirmed macOS 27 bug or regression.
Apple's documentation for Desktop Stacks is here:
Get to know your desktop, menu bar, and Dock — Apple Support