### 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 left_order = p_left->owner ? p_left->owner->data.process_thread_group_order : 0;
	int right_order = p_right->owner ? p_right->owner->data.process_thread_group_order : 0;
	if (left_order == right_order) {
		int left_threaded = p_left->owner != nullptr && p_left->owner->data.process_thread_group == Node::PROCESS_THREAD_GROUP_SUB_THREAD ? 0 : 1;
		int right_threaded = p_right->owner != nullptr && p_right->owner->data.process_thread_group == Node::PROCESS_THREAD_GROUP_SUB_THREAD ? 0 : 1;
		return left_threaded < right_threaded;
	} else {
		return left_order < right_order;
	}
}
void SceneTree::_remove_process_group(Node *p_node) {
	_THREAD_SAFE_METHOD_
	ProcessGroup *pg = (ProcessGroup *)p_node->data.process_group;
	ERR_FAIL_COND(!pg);
	ERR_FAIL_COND(pg->removed);
	pg->removed = true;
	pg->owner = nullptr;
	p_node->data.process_group = nullptr;
	process_groups_dirty = true;
}
void SceneTree::_add_process_group(Node *p_node) {
	_THREAD_SAFE_METHOD_
	ERR_FAIL_COND(!p_node);
	ProcessGroup *pg = memnew(ProcessGroup);
	pg->owner = p_node;
	p_node->data.process_group = pg;
	process_groups.push_back(pg);
	process_groups_dirty = true;
}
void SceneTree::_remove_node_from_process_group(Node *p_node, Node *p_owner) {