### 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)

Ref<Theme> theme = EditorNode::get_singleton()->get_editor_theme();
	ERR_FAIL_COND_V(theme.is_null(), Ref<ImageTexture>());
	if (EditorNode::get_singleton()->get_main_screen_control()->is_layout_rtl()) {
		return theme->get_icon(SNAME("PlayBackwards"), EditorStringName(EditorIcons));
	} else {
		return theme->get_icon(SNAME("Play"), EditorStringName(EditorIcons));
	}
}
String EditorExportPlatform::find_export_template(String template_file_name, String *err) const {
	String current_version = VERSION_FULL_CONFIG;
	String template_path = EditorPaths::get_singleton()->get_export_templates_dir().path_join(current_version).path_join(template_file_name);
	if (FileAccess::exists(template_path)) {
		return template_path;
	}
	
	if (err) {
		*err += TTR("No export template found at the expected path:") + "\n" + template_path + "\n";
	}
	return String();
}
bool EditorExportPlatform::exists_export_template(String template_file_name, String *err) const {