### Godot version

4.0.2

### System information

Windows 10, GPU with no Vulkan support

### Issue description

My GPU doesn't support Vulkan. I run Godot 4 with "--rendering-driver opengl3" argument.

I have tried to open my Godot 3 project from Godot 4 Project Manager. It asked me whether I want my project to be converted. I pressed "full conversion". Very shortly it gave me a error that it can't run Vulkan app and that I should use "--rendering-driver opengl3", which I already run. After it closed Project Manager for me and I reopened it, I could find the project in the Project Manager list and then open it. There was no error. However the convertion was minimal, just enough to run it in Godot 4. I believe that it didn't run a full auto-convertion.
This always happens when I repeat these steps.

Next I have used command line and launched Godot 4 from it with "--convert-3to4" argument, doing it on second copy of the same Godot 3 project. (Had to increase limits for it to convert everything.) The process went smoothly and final project could be opened without error as well. However it had a more full conversion in comparison. This makes me believe that conversion process fails in Project Manager method by not respecting "--rendering-driver opengl3" argument in the process (I don't know how it actually looks as I can't run it).

P.S. I also run Godot 4 with "--single-window" argument, as it glitches out for me otherwise. I don't know if it has an impact on my experience, but worth mentioning.

Bonus:
I have a suspicion that this issue 
https://github.com/godotengine/godot/issues/74159
is related to this one and is actually a result of a failed conversion.

### Steps to reproduce

Get GPU with no Vulkan support and try to convert any Godot 3 project from Project Manager while running with "--rendering-driver opengl3".

### Minimal reproduction project

Any Godot 3 project, it just has to error on attempt.
for (const PropertyInfo &pi : edor_settings_properties) {
			if (pi.name == "interface/edor/edor_language") {
				edor_languages = pi.hint_string.spl(",");
				break;
			}
		}
		String current_lang = EDITOR_GET("interface/edor/edor_language");
		language_btn->set_text(current_lang);
		for (int i = 0; i < edor_languages.size(); i++) {
			String lang = edor_languages[i];
			String lang_name = TranslationServer::get_singleton()->get_locale_name(lang);
			language_btn->add_em(vformat("[%s] %s", lang, lang_name), i);
			language_btn->set_em_metadata(i, lang);
			if (current_lang == lang) {
				language_btn->select(i);
			}
		}
		settings_hb->add_child(language_btn);
		center_box->add_child(settings_hb);
	}
	if (AssetLibraryEdorPlugin::is_available()) {
		asset_library = memnew(EdorAssetLibrary(true));
		asset_library->set_name(TTR("Asset Library Projects"));
		tabs->add_child(asset_library);
		asset_library->connect("install_asset", callable_mp(this, &ProjectManager::_install_project));
	} else {
		print_verbose("Asset Library not available (due to using Web edor, or SSL support disabled).");
	}
	{
		// Dialogs