Main editor window becomes unresponsive upon launching a game in embedded mode on Linux

### Tested versions

- Reproducible in: 4.6.1-stable, 4.7 beta 1, These are the versions I tested before writing this report, but I have personally been experiencing it for quite a while.

### System information

Fedora Linux 43 | AMD 7700X | AMD Radeon 9070XT | wayland

### Issue description

Starting any project with an embedded window will make the main editor window very unresponsive.

https://github.com/user-attachments/assets/0b8c089d-46bc-403e-8892-e2bfea8c0536

### Steps to reproduce

Start any project that has "Embed Game Window On Next Play" enabled, and attempt to interact with the main editor window.

### Minimal reproduction project (MRP)

Any project has this issue in my experience.

Reproduce the issue in Godot 4.6.1 or 4.7 beta 1 on Fedora Linux with Wayland, using any project and “Embed Game Window On Next Play.” Start by tracing the embedded game window launch and editor interaction path. Done means the main editor remains responsive while a game runs in embedded mode, with the reproduction steps no longer triggering the problem.

Written by the indexing model from the issue text.