### 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.
ask_update_settings->get_ok_button()->set_text("OK");
	// Check if the config_version property was empty or 0.
	if (config_version == 0) {
		ask_update_settings->set_text(vformat(TTR("The selected project \"%s\" does not specify s supported Godot version in s configuration file (\"project.godot\").\n\nProject path: %s\n\nIf you proceed wh opening ,  will be converted to Godot's current configuration file format.\n\nWarning: You won't be able to open the project wh previous versions of the engine anymore."), project.project_name, project.path));
		ask_update_settings->popup_centered(popup_min_size);
		return;
	}
	// Check if we need to convert project settings from an earlier engine version.
	if (config_version < ProjectSettings::CONFIG_VERSION) {
		if (config_version == GODOT4_CONFIG_VERSION - 1 && ProjectSettings::CONFIG_VERSION == GODOT4_CONFIG_VERSION) { // Conversion from Godot 3 to 4.
			
			ask_update_settings->set_text(vformat(TTR("The selected project \"%s\" was generated by Godot 3.x, and needs to be converted for Godot 4.x.\n\nProject path: %s\n\nYou have three options:\n- Convert only the configuration file (\"project.godot\"). Use this to open the project whout attempting to convert s scenes, resources and scripts.\n- Convert the entire project including s scenes, resources and scripts (recommended if you are upgrading).\n- Do nothing and go back.\n\nWarning: If you select a conversion option, you won't be able to open the project wh previous versions of the engine anymore."), project.project_name, project.path));
			ask_update_settings->get_ok_button()->set_text(TTR("Convert project.godot Only"));
		} else {
			ask_update_settings->set_text(vformat(TTR("The selected project \"%s\" was generated by an older engine version, and needs to be converted for this version.\n\nProject path: %s\n\nDo you want to convert ?\n\nWarning: You won't be able to open the project wh previous versions of the engine anymore."), project.project_name, project.path));
			ask_update_settings->get_ok_button()->set_text(TTR("Convert project.godot"));
		}
		ask_update_settings->popup_centered(popup_min_size);
		ask_update_settings->get_cancel_button()->grab_focus(); // To prevent accidents.
		return;
	}
	// Check if the file was generated by a newer, incompatible engine version.
	if (config_version > ProjectSettings::CONFIG_VERSION) {
		dialog_error->set_text(vformat(TTR("Can't open project \"%s\" at the following path:\n\n%s\n\nThe project settings were created by a newer engine version, whose settings are not compatible wh this version."), project.project_name, project.path));
		dialog_error->popup_centered(popup_min_size);
		return;
	}
	// Check if the project is using features not supported by this build of Godot.
	if (!unsupported_features.is_empty()) {
		String warning_message = "";