Hi Joao Silva,
Based on your testing, this appears to be a compatibility issue because the controller is still detected by Windows and works in joy.cpl/Xbox app, while games stop recognizing it after KB5127936 is installed. Uninstalling the update restoring functionality further narrows down the trigger.
As a temporary workaround, you can stay on the working configuration and pause updates for a short period. I also suggest submitting a report through Feedback Hub (Windows + F), including your Windows build, KB5127936, Xbox 360 Wireless Receiver, and the affected games.
Please note that this is a user-to-user support forum. Moderators, contributors including external Microsoft employees cannot directly intervene in Microsoft product features or access back-end systems. Our role is limited to providing technical guidance on reported issues, requests, or ideas.
That information will be useful for further investigation, and hopefully an upcoming update will restore compatibility.