### 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)

main_vb->add_child(export_texture_format_error);
	export_texture_format_error->hide();
	export_texture_format_error->connect("texture_format_enabled", callable_mp(this, &ProjectExportDialog::_update_current_preset));
	export_error = memnew(Label);
	main_vb->add_child(export_error);
	export_error->hide();
	export_error->add_theme_color_override("font_color", EditorNode::get_singleton()->get_editor_theme()->get_color(SNAME("error_color"), EditorStringName(Editor)));
	export_warning = memnew(Label);
	main_vb->add_child(export_warning);
	export_warning->hide();
	export_warning->add_theme_color_override("font_color", EditorNode::get_singleton()->get_editor_theme()->get_color(SNAME("warning_color"), EditorStringName(Editor)));
	export_templates_error = memnew(HBoxContainer);
	main_vb->add_child(export_templates_error);
	export_templates_error->hide();
	Label *export_error2 = memnew(Label);
	export_templates_error->add_child(export_error2);
	export_error2->add_theme_color_override("font_color", EditorNode::get_singleton()->get_editor_theme()->get_color(SNAME("error_color"), EditorStringName(Editor)));
	export_error2->set_text(String::utf8("•  ") + TTR("Export templates for this platform are missing:") + " ");
	result_dialog = memnew(AcceptDialog);
	result_dialog->set_title(TTR("Project Export"));
	result_dialog_log = memnew(RichTextLabel);
	result_dialog_log->set_custom_minimum_size(Size2(300, 80) * EDSCALE);
	result_dialog->add_child(result_dialog_log);
	main_vb->add_child(result_dialog);
	result_dialog->hide();
	LinkButton *download_templates = memnew(LinkButton);
	download_templates->set_text(TTR("Manage Export Templates"));
	download_templates->set_v_size_flags(Control::SIZE_SHRINK_CENTER);
	export_templates_error->add_child(download_templates);
	download_templates->connect("pressed", callable_mp(this, &ProjectExportDialog::_open_export_template_manager));