### 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_
	ProcessGroup *pg = p_owner ? (ProcessGroup *)p_owner->data.process_group : &default_process_group;
	if (p_node->is_processing() || p_node->is_processing_internal()) {
		bool found = pg->nodes.erase(p_node);
		ERR_FAIL_COND(!found);
	}
	if (p_node->is_physics_processing() || p_node->is_physics_processing_internal()) {
		bool found = pg->physics_nodes.erase(p_node);
		ERR_FAIL_COND(!found);
	}
}
void SceneTree::_add_node_to_process_group(Node *p_node, Node *p_owner) {
	_THREAD_SAFE_METHOD_
	ProcessGroup *pg = p_owner ? (ProcessGroup *)p_owner->data.process_group : &default_process_group;
	if (p_node->is_processing() || p_node->is_processing_internal()) {
		pg->nodes.push_back(p_node);
		pg->node_order_dirty = true;
	}
	if (p_node->is_physics_processing() || p_node->is_physics_processing_internal()) {
		pg->physics_nodes.push_back(p_node);
		pg->physics_node_order_dirty = true;
	}
}
void SceneTree::_call_input_pause(const StringName &p_group, CallInputType p_call_type, const Ref<InputEvent> &p_input, Viewport *p_viewport) {