### 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)

};
	
	int diamond_faces[8 * 3] = {
		0, 2, 4,
		0, 3, 4,
		1, 2, 4,
		1, 3, 4,
		0, 2, 5,
		0, 3, 5,
		1, 2, 5,
		1, 3, 5,
	};
	
	Vector<int> indices;
	for (int i = 0; i < 8 * 3; i++) {
		indices.push_back(diamond_faces[i]);
	}
	Vector<Vector3> vertices;
	for (int i = 0; i < 6; i++) {
		vertices.push_back(diamond[i] * 0.1);
	}
	Array arr;
	arr.resize(Mesh::ARRAY_MAX);
	arr[Mesh::ARRAY_VERTEX] = vertices;
	arr[Mesh::ARRAY_INDEX] = indices;
	debug_contact_mesh->add_surface_from_arrays(Mesh::PRIMITIVE_TRIANGLES, arr);
	debug_contact_mesh->surface_set_material(0, mat);
	return debug_contact_mesh;
}
void SceneTree::set_pause(bool p_enabled) {