### 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.
tag_vb->add_child(tag_error);
		create_tag_btn = memnew(Button);
		all_tags->add_child(create_tag_btn);
		create_tag_btn->connect("pressed", callable_mp((Window *)create_tag_dialog, &Window::popup_centered).bind(Vector2i(500, 0) * EDSCALE));
	}
	_project_list->migrate_config();
	_load_recent_projects();
	Ref<DirAccess> dir_access = DirAccess::create(DirAccess::AccessType::ACCESS_FILESYSTEM);
	String default_project_path = EDITOR_GET("filesystem/directories/default_project_path");
	if (!dir_access->dir_exists(default_project_path)) {
		Error error = dir_access->make_dir_recursive(default_project_path);
		if (error != OK) {
			ERR_PRINT("Could not create default project directory at: " + default_project_path);
		}
	}
	String autoscan_path = EDITOR_GET("filesystem/directories/autoscan_project_path");
	if (!autoscan_path.is_empty()) {
		if (dir_access->dir_exists(autoscan_path)) {
			_scan_begin(autoscan_path);
		} else {
			Error error = dir_access->make_dir_recursive(autoscan_path);
			if (error != OK) {
				ERR_PRINT("Could not create project autoscan directory at: " + autoscan_path);
			}
		}
	}
	SceneTree::get_singleton()->get_root()->connect("files_dropped", callable_mp(this, &ProjectManager::_files_dropped));
	// Define a minimum window size to prevent UI elements from overlapping or being cut off.
	Window *w = Object::cast_to<Window>(SceneTree::get_singleton()->get_root());
	if (w) {