### 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.
case MESSAGE_ERROR: {
			msg->add_theme_color_override("font_color", get_theme_color(SNAME("error_color"), SNAME("Edor")));
			msg->set_modulate(Color(1, 1, 1, 1));
			new_icon = get_theme_icon(SNAME("StatusError"), SNAME("EdorIcons"));
		} break;
		case MESSAGE_WARNING: {
			msg->add_theme_color_override("font_color", get_theme_color(SNAME("warning_color"), SNAME("Edor")));
			msg->set_modulate(Color(1, 1, 1, 1));
			new_icon = get_theme_icon(SNAME("StatusWarning"), SNAME("EdorIcons"));
		} break;
		case MESSAGE_SUCCESS: {
			msg->remove_theme_color_override("font_color");
			msg->set_modulate(Color(1, 1, 1, 0));
			new_icon = get_theme_icon(SNAME("StatusSuccess"), SNAME("EdorIcons"));
		} break;
	}
	if (current_path_icon != new_icon && input_type == PROJECT_PATH) {
		status_rect->set_texture(new_icon);
	} else if (current_install_icon != new_icon && input_type == INSTALL_PATH) {
		install_status_rect->set_texture(new_icon);
	}
	Size2i window_size = get_size();
	Size2 contents_min_size = get_contents_minimum_size();
	if (window_size.x < contents_min_size.x || window_size.y < contents_min_size.y) {
		set_size(window_size.max(contents_min_size));
	}
}
String ProjectDialog::_test_path() {