### 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)

r_error.error = Callable::CallError::CALL_OK;
	ERR_FAIL_COND(p_argcount < 3);
	ERR_FAIL_COND(!p_args[0]->is_num());
	ERR_FAIL_COND(p_args[1]->get_type() != Variant::STRING_NAME && p_args[1]->get_type() != Variant::STRING);
	ERR_FAIL_COND(p_args[2]->get_type() != Variant::STRING_NAME && p_args[2]->get_type() != Variant::STRING);
	int flags = *p_args[0];
	StringName group = *p_args[1];
	StringName method = *p_args[2];
	call_group_flagsp(flags, group, method, p_args + 3, p_argcount - 3);
}
void SceneTree::_call_group(const Variant **p_args, int p_argcount, Callable::CallError &r_error) {
	r_error.error = Callable::CallError::CALL_OK;
	ERR_FAIL_COND(p_argcount < 2);
	ERR_FAIL_COND(p_args[0]->get_type() != Variant::STRING_NAME && p_args[0]->get_type() != Variant::STRING);
	ERR_FAIL_COND(p_args[1]->get_type() != Variant::STRING_NAME && p_args[1]->get_type() != Variant::STRING);
	StringName group = *p_args[0];
	StringName method = *p_args[1];
	call_group_flagsp(GROUP_CALL_DEFAULT, group, method, p_args + 2, p_argcount - 2);
}
int64_t SceneTree::get_frame() const {