### 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)

List<String> list;
	list.push_back("html");
	return list;
}
Error EditorExportPlatformWeb::export_project(const Ref<EditorExportPreset> &p_preset, bool p_debug, const String &p_path, int p_flags) {
	ExportNotifier notifier(*this, p_preset, p_debug, p_path, p_flags);
	const String custom_debug = p_preset->get("custom_template/debug");
	const String custom_release = p_preset->get("custom_template/release");
	const String custom_html = p_preset->get("html/custom_html_shell");
	const bool export_icon = p_preset->get("html/export_icon");
	const bool pwa = p_preset->get("progressive_web_app/enabled");
	const String base_dir = p_path.get_base_dir();
	const String base_path = p_path.get_basename();
	const String base_name = p_path.get_file().get_basename();
	
	String template_path = p_debug ? custom_debug : custom_release;
	template_path = template_path.strip_edges();
	if (template_path.is_empty()) {
		bool extensions = (bool)p_preset->get("variant/extensions_support");
		template_path = find_export_template(_get_template_name(extensions, p_debug));
	}
	if (!DirAccess::exists(base_dir)) {
		return ERR_FILE_BAD_PATH;