### 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.
tree_vb->add_child(rename_btn);
		manage_tags_btn = memnew(Button);
		manage_tags_btn->set_text(TTR("Manage Tags"));
		tree_vb->add_child(manage_tags_btn);
		erase_btn = memnew(Button);
		erase_btn->set_text(TTR("Remove"));
		erase_btn->set_shortcut(ED_SHORTCUT("project_manager/remove_project", TTR("Remove Project"), Key::KEY_DELETE));
		erase_btn->connect("pressed", callable_mp(this, &ProjectManager::_erase_project));
		tree_vb->add_child(erase_btn);
		erase_missing_btn = memnew(Button);
		erase_missing_btn->set_text(TTR("Remove Missing"));
		erase_missing_btn->connect("pressed", callable_mp(this, &ProjectManager::_erase_missing_projects));
		tree_vb->add_child(erase_missing_btn);
		tree_vb->add_spacer();
		about_btn = memnew(Button);
		about_btn->set_text(TTR("About"));
		about_btn->connect("pressed", callable_mp(this, &ProjectManager::_show_about));
		tree_vb->add_child(about_btn);
	}
	{
		// Version info and language options
		settings_hb = memnew(HBoxContainer);
		settings_hb->set_alignment(BoxContainer::ALIGNMENT_END);
		settings_hb->set_h_grow_direction(Control::GROW_DIRECTION_BEGIN);
		settings_hb->set_anchors_and_offsets_preset(Control::PRESET_TOP_RIGHT);
		// A VBoxContainer that contains a dummy Control node to adjust the LinkButton's vertical posion.
		VBoxContainer *spacer_vb = memnew(VBoxContainer);
		settings_hb->add_child(spacer_vb);
		Control *v_spacer = memnew(Control);
		spacer_vb->add_child(v_spacer);