Not sure if Windows has that but I believe on macOS what happens is the app tries to record the screen, and if it fails macOS blocks the request and opens the security settings to enable the permission, and you have restart the whole application for the permission to take.
What's done for Wayland is the portal system: applications can use portals to request access to specific things like screen recording, the DE does what it needs to do and it starts feeding the data to the application through the portal. It's working fairly well, I haven't had issues with those in a while. The application just requests what it wants, and the DE prompts the user (or auto accept the request) optionally remembering the choice as well.
Generally the solution for X11 problems is to implement a modern API for it in either Wayland or as a portal. Which breaks old stuff, but once updated it works fine.
The main obstacle is getting Gnome to agree to the protocols.