How to target a specific custom post type post and its all children and grandchildren?

I guess you want to check if the active page is a) of your custom post type and b) one of the ancestors is your parameter.

First of all, i would advise you to keep on using the ID, as slugs and titles can be non-unique, so one slug could be different posts.

Second, you can use wordpress’ own get_post_ancestors-function. Use it like this:

function wpse365743_has_ancestor($postid,$ancestor_id){
  //get all parents and ancestors of $postid
  $ancestors = get_post_ancestors($postid);
  //add in the current post id
  $ancestors = array_merge(array($postid),$ancestors);
  //is the searched ancestor id in the ancestors?
  return in_array($ancestor_id,$ancestors); 
}

You can now use this anywhere on your page like this:

if(('my_post_type_name' == get_post_type($check_post_id)) && wpse365743_has_ancestor($check_post_id,$searched_ancestor_id)){
   //do something
}

Happy Coding!