### 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)

#define CLEANUP_AND_RETURN(m_err)                       \
	{                                                   \
		if (da->file_exists(basepath + ".zip")) {       \
			da->remove(basepath + ".zip");              \
		}                                               \
		if (da->file_exists(basepath + "_start.ps1")) { \
			da->remove(basepath + "_start.ps1");        \
		}                                               \
		if (da->file_exists(basepath + "_clean.ps1")) { \
			da->remove(basepath + "_clean.ps1");        \
		}                                               \
		return m_err;                                   \
	}                                                   \
	((void)0)
	if (ep.step(TTR("Exporting project..."), 1)) {
		return ERR_SKIP;
	}
	Error err = export_project(p_preset, true, basepath + ".zip", p_debug_flags);
	if (err != OK) {
		DirAccess::remove_file_or_error(basepath + ".zip");
		return err;
	}
	String cmd_args;
	{
		Vector<String> cmd_args_list;
		gen_debug_flags(cmd_args_list, p_debug_flags);
		for (int i = 0; i < cmd_args_list.size(); i++) {
			if (i != 0) {
				cmd_args += " ";
			}