Unable to rename a project after quitting to project list

[carvalho-paulo]

Tested versions

v4.4.beta2.official [a013481]
v4.4.beta1.official [d33da79]

System information

Windows 11 - Godot v4.4.beta2

Issue description

After quitting to the project list, it is not possible to rename the project we have just quit from.

Steps to reproduce

1. Open any project
2. Quit to Project List (through the Project menu or the shortcut Ctrl+Shift+Q)
3. Select the project you just quit from and click on Rename
4. Note that you won't be able to rename the project and the following message is shown in red:

Creating a project at the engine's working directory or executable directory is not allowed, as it would prevent the project manager from starting.

Minimal reproduction project (MRP)

Can be reproduced with any project.

[Saulo-de-Souza]

It seems to have something to do with #101890, as it seems that underneath it all, the last closed project is still open.

[matheusmdx]

Is a regression between 4.4 dev 3 and 4.4 dev 4, bisecting rn.

[matheusmdx]

Bisected to #94880, @KoBeWi