the permission for screen access is only for apps that failed to start using the new API where the OS lets the user select a window/screen they want to share.
Not sure if it is broken, but I took the opportunity when upgrading today to say No and just disable it. The product is EOL anyway, there are a few upstarts in the open source world, and I am hoping the best for Apples on-device SLMs (or oss ones) mooting the requirement for OpenAI.
Of all the apps I use that record my screen, literally all of them have updated to this new API ages ago. That includes Discord, Slack, Teams, Zoom, FaceTime, OBS, and Safari. I want to say that Firefox has as well but can't remember for certain.