### Tested versions

Godot Engine v4.2.stable.official.46dc27791

### System information

Windows 10

### Issue description

When exporting to web, Godot fails with this error in the console:

> scene/main/window.cpp:884 - Attempting to make child window exclusive, but the parent window already has another exclusive child. This window: /root/@<!-- -->EditorNode@<!-- -->17120/@<!-- -->Panel@<!-- -->13/@<!-- -->ProjectExportDialog@<!-- -->559/@<!-- -->VBoxContainer@<!-- -->137/@<!-- -->AcceptDialog@<!-- -->411, parent window: /root/@<!-- -->EditorNode@<!-- -->17120/@<!-- -->Panel@<!-- -->13/@<!-- -->ProjectExportDialog@<!-- -->559, current exclusive child window: /root/@<!-- -->EditorNode@<!-- -->17120/@<!-- -->Panel@<!-- -->13/@<!-- -->ProjectExportDialog@<!-- -->559/@<!-- -->ProgressDialog@<!-- -->17444

![a](https://github.com/godotengine/godot/assets/2000051/bf9d22aa-d3fe-4e11-bff6-c387e740fe90)

This happens for newly created empty projects.


### Steps to reproduce

1. Open the attached project (or create a new project).
2. Project->Export->Export All
3. Observe the console error.  No output files are generated.


### Minimal reproduction project (MRP)

[export_test.zip](https://github.com/godotengine/godot/files/13574810/export_test.zip)

if (i > 0) {
					error += "\n";
				}
				error += " - " + items[i];
			}
			export_error->set_text(error);
			export_error->show();
		} else {
			export_error->hide();
		}
		if (needs_templates) {
			export_templates_error->show();
		} else {
			export_templates_error->hide();
		}
		export_warning->hide();
		export_button->set_disabled(true);
	} else {
		if (error != String()) {
			Vector<String> items = error.split("\n", false);
			error = "";
			for (int i = 0; i < items.size(); i++) {
				if (i > 0) {
					error += "\n";
				}
				error += " - " + items[i];
			}
			export_warning->set_text(error);
			export_warning->show();
		} else {