### 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_paths_material.is_valid()) {
		return debug_paths_material;
	}
	Ref<StandardMaterial3D> _debug_material = Ref<StandardMaterial3D>(memnew(StandardMaterial3D));
	_debug_material->set_shading_mode(StandardMaterial3D::SHADING_MODE_UNSHADED);
	_debug_material->set_transparency(StandardMaterial3D::TRANSPARENCY_ALPHA);
	_debug_material->set_flag(StandardMaterial3D::FLAG_SRGB_VERTEX_COLOR, true);
	_debug_material->set_flag(StandardMaterial3D::FLAG_ALBEDO_FROM_VERTEX_COLOR, true);
	_debug_material->set_albedo(get_debug_paths_color());
	debug_paths_material = _debug_material;
	return debug_paths_material;
}
Ref<Material> SceneTree::get_debug_collision_material() {
	_THREAD_SAFE_METHOD_
	if (collision_material.is_valid()) {
		return collision_material;
	}
	Ref<StandardMaterial3D> line_material = Ref<StandardMaterial3D>(memnew(StandardMaterial3D));
	line_material->set_shading_mode(StandardMaterial3D::SHADING_MODE_UNSHADED);
	line_material->set_transparency(StandardMaterial3D::TRANSPARENCY_ALPHA);
	line_material->set_flag(StandardMaterial3D::FLAG_SRGB_VERTEX_COLOR, true);
	line_material->set_flag(StandardMaterial3D::FLAG_ALBEDO_FROM_VERTEX_COLOR, true);
	line_material->set_albedo(get_debug_collisions_color());
	collision_material = line_material;
	return collision_material;
}
Ref<ArrayMesh> SceneTree::get_debug_contact_mesh() {