### 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)

return debug_avoidance_hint;
}
#endif
void SceneTree::set_debug_collisions_color(const Color &p_color) {
	debug_collisions_color = p_color;
}
Color SceneTree::get_debug_collisions_color() const {
	return debug_collisions_color;
}
void SceneTree::set_debug_collision_contact_color(const Color &p_color) {
	debug_collision_contact_color = p_color;
}
Color SceneTree::get_debug_collision_contact_color() const {
	return debug_collision_contact_color;
}
void SceneTree::set_debug_paths_color(const Color &p_color) {
	debug_paths_color = p_color;
}
Color SceneTree::get_debug_paths_color() const {
	return debug_paths_color;
}
void SceneTree::set_debug_paths_width(float p_width) {
	debug_paths_width = p_width;
}
float SceneTree::get_debug_paths_width() const {
	return debug_paths_width;
}
Ref<Material> SceneTree::get_debug_paths_material() {