### Godot version

v4.1.beta1.mono.official [828ec2c5d]

### System information

Godot v4.1.beta1.mono - Windows 10.0.22621 - Vulkan (Forward+) - dedicated NVIDIA GeForce RTX 4070 Ti (NVIDIA; 31.0.15.3598) - 13th Gen Intel(R) Core(TM) i5-13600K (20 Threads)

### Issue description

After the recent refactoring (https://github.com/godotengine/godot/pull/75901), Node process priority seems to be unimplemented. `ComparatorWithPriority` is no longer used anywhere in the engine code.

Nodes behave as if they all had the same priority, and running with `--single-threaded-scene` makes no difference.

### Steps to reproduce

Run the MRP, look at the order in which the nodes print their names.

In Godot 4.0.3: First, Default, Last
In Godot 4.1-beta1: Default, First, Last

### Minimal reproduction project

[ProcessPriorityBug.zip](https://github.com/godotengine/godot/files/11693028/ProcessPriorityBug.zip)

_THREAD_SAFE_METHOD_
	if (debug_contact_mesh.is_valid()) {
		return debug_contact_mesh;
	}
	debug_contact_mesh = Ref<ArrayMesh>(memnew(ArrayMesh));
	Ref<StandardMaterial3D> mat = Ref<StandardMaterial3D>(memnew(StandardMaterial3D));
	mat->set_shading_mode(StandardMaterial3D::SHADING_MODE_UNSHADED);
	mat->set_transparency(StandardMaterial3D::TRANSPARENCY_ALPHA);
	mat->set_flag(StandardMaterial3D::FLAG_SRGB_VERTEX_COLOR, true);
	mat->set_flag(StandardMaterial3D::FLAG_ALBEDO_FROM_VERTEX_COLOR, true);
	mat->set_albedo(get_debug_collision_contact_color());
	Vector3 diamond[6] = {
		Vector3(-1, 0, 0),
		Vector3(1, 0, 0),
		Vector3(0, -1, 0),
		Vector3(0, 1, 0),
		Vector3(0, 0, -1),
		Vector3(0, 0, 1)