### 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 (EditorNode::get_singleton()) {
		server.instantiate();
		server_thread.start(_server_thread_poll, this);
#ifdef MODULE_SVG_ENABLED
		Ref<Image> img = memnew(Image);
		const bool upsample = !Math::is_equal_approx(Math::round(EDSCALE), EDSCALE);
		ImageLoaderSVG::create_image_from_string(img, _web_logo_svg, EDSCALE, upsample, false);
		logo = ImageTexture::create_from_image(img);
		ImageLoaderSVG::create_image_from_string(img, _web_run_icon_svg, EDSCALE, upsample, false);
		run_icon = ImageTexture::create_from_image(img);
#endif
		Ref<Theme> theme = EditorNode::get_singleton()->get_editor_theme();
		if (theme.is_valid()) {
			stop_icon = theme->get_icon(SNAME("Stop"), EditorStringName(EditorIcons));
		} else {
			stop_icon.instantiate();
		}
	}
}
EditorExportPlatformWeb::~EditorExportPlatformWeb() {
	if (server.is_valid()) {
		server->stop();
	}
	server_quit = true;
	if (server_thread.is_started()) {
		server_thread.wait_to_finish();
	}
}