### 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<FileAccess> f = FileAccess::open(p_path, FileAccess::WRITE);
	if (f.is_null()) {
		add_message(EXPORT_MESSAGE_ERROR, TTR("Export"), vformat(TTR("Could not write file: \"%s\"."), p_path));
		return ERR_FILE_CANT_WRITE;
	}
	f->store_buffer(p_content, p_size);
	return OK;
}
void EditorExportPlatformWeb::_replace_strings(HashMap<String, String> p_replaces, Vector<uint8_t> &r_template) {
	String str_template = String::utf8(reinterpret_cast<const char *>(r_template.ptr()), r_template.size());
	String out;
	Vector<String> lines = str_template.split("\n");
	for (int i = 0; i < lines.size(); i++) {
		String current_line = lines[i];
		for (const KeyValue<String, String> &E : p_replaces) {
			current_line = current_line.replace(E.key, E.value);
		}
		out += current_line + "\n";
	}
	CharString cs = out.utf8();
	r_template.resize(cs.length());
	for (int i = 0; i < cs.length(); i++) {
		r_template.write[i] = cs[i];
	}
}
void EditorExportPlatformWeb::_fix_html(Vector<uint8_t> &p_html, const Ref<EditorExportPreset> &p_preset, const String &p_name, bool p_debug, int p_flags, const Vector<SharedObject> p_shared_objects, const Dictionary &p_file_sizes) {