### 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)

quit_on_go_back = p_enable;
}
#ifdef TOOLS_ENABLED
bool SceneTree::is_node_being_edited(const Node *p_node) const {
	return Engine::get_singleton()->is_editor_hint() && edited_scene_root && (edited_scene_root->is_ancestor_of(p_node) || edited_scene_root == p_node);
}
#endif
#ifdef DEBUG_ENABLED
void SceneTree::set_debug_collisions_hint(bool p_enabled) {
	debug_collisions_hint = p_enabled;
}
bool SceneTree::is_debugging_collisions_hint() const {
	return debug_collisions_hint;
}
void SceneTree::set_debug_paths_hint(bool p_enabled) {
	debug_paths_hint = p_enabled;
}
bool SceneTree::is_debugging_paths_hint() const {
	return debug_paths_hint;
}
void SceneTree::set_debug_navigation_hint(bool p_enabled) {
	debug_navigation_hint = p_enabled;
}
bool SceneTree::is_debugging_navigation_hint() const {
	return debug_navigation_hint;
}
void SceneTree::set_debug_avoidance_hint(bool p_enabled) {
	debug_avoidance_hint = p_enabled;
}
bool SceneTree::is_debugging_avoidance_hint() const {