### 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.
if (!_search_term.is_empty()) {
			String search_path;
			if (search_term.contains("/")) {
				// Search path will match the whole path
				search_path = em.path;
			} else {
				// Search path will only match the last path component to make searching more strict
				search_path = em.path.get_file();
			}
			bool missing_tags = false;
			for (const String &tag : tags) {
				if (!em.tags.has(tag)) {
					missing_tags = true;
					break;
				}
			}
			// When searching, display projects whose name or path contain the search term and whose tags match the searched tags.
			em_visible = !missing_tags && (search_term.is_empty() || em.project_name.findn(search_term) != -1 || search_path.findn(search_term) != -1);
		}
		em.control->set_visible(em_visible);
	}
	for size(); ++i) {
		Item &em = _projects.wre[i];
		em.control->get_parent()->move_child(em.control, i);
	}
	// Rewind the coroutine because order of projects changed
	_update_icons_async();
	update_dock_menu();
}
const HashSet<String> &ProjectList::get_selected_project_keys() const {