### 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)

node_threading_disabled = p_disable;
}
SceneTree::SceneTree() {
	if (singleton == nullptr) {
		singleton = this;
	}
	debug_collisions_color = GLOBAL_DEF("debug/shapes/collision/shape_color", Color(0.0, 0.6, 0.7, 0.42));
	debug_collision_contact_color = GLOBAL_DEF("debug/shapes/collision/contact_color", Color(1.0, 0.2, 0.1, 0.8));
	debug_paths_color = GLOBAL_DEF("debug/shapes/paths/geometry_color", Color(0.1, 1.0, 0.7, 0.4));
	debug_paths_width = GLOBAL_DEF("debug/shapes/paths/geometry_width", 2.0);
	collision_debug_contacts = GLOBAL_DEF(PropertyInfo(Variant::INT, "debug/shapes/collision/max_contacts_displayed", PROPERTY_HINT_RANGE, "0,20000,1"), 10000);
	GLOBAL_DEF("debug/shapes/collision/draw_2d_outlines", true);
	process_group_call_queue_allocator = memnew(CallQueue::Allocator(64));
	Math::randomize();
	
	root = memnew(Window);
	root->set_min_size(Size2i(64, 64)); 
	root->set_process_mode(Node::PROCESS_MODE_PAUSABLE);
	root->set_name("root");
	root->set_title(GLOBAL_GET("application/config/name"));
	if (Engine::get_singleton()->is_editor_hint()) {
		root->set_wrap_controls(true);
	}
#ifndef _3D_DISABLED
	if (!root->get_world_3d().is_valid()) {
		root->set_world_3d(Ref<World3D>(memnew(World3D)));