### 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)

sec_info->set_text(TTR("Note: Encryption key needs to be stored in the binary,\nyou need to build the export templates from source."));
	sec_vb->add_child(sec_info);
	LinkButton *sec_more_info = memnew(LinkButton);
	sec_more_info->set_text(TTR("More Info..."));
	sec_more_info->connect("pressed", callable_mp(this, &ProjectExportDialog::_open_key_help_link));
	sec_vb->add_child(sec_more_info);
	sections->connect("tab_changed", callable_mp(this, &ProjectExportDialog::_tab_changed));
	// 
	name->set_editable(false);
	export_path->hide();
	runnable->set_disabled(true);
	duplicate_preset->set_disabled(true);
	delete_preset->set_disabled(true);
	script_key_error->hide();
	sections->hide();
	parameters->edit(nullptr);
	// 
	delete_confirm = memnew(ConfirmationDialog);
	add_child(delete_confirm);
	delete_confirm->set_ok_button_text(TTR("Delete"));
	delete_confirm->connect("confirmed", callable_mp(this, &ProjectExportDialog::_delete_preset_confirm));
	// 
	set_cancel_button_text(TTR("Close"));
	set_ok_button_text(TTR("Export PCK/ZIP..."));
	get_ok_button()->set_disabled(true);
#ifdef ANDROID_ENABLED
	export_button = memnew(Button);
	export_button->hide();
#else
	export_button = add_button(TTR("Export Project..."), !DisplayServer::get_singleton()->get_swap_cancel_ok(), "export");