Yes this would be a good idea to use the get_terms but we will run into an issue with using this method.
The problem would be that in order for us to get the correct information to be display we must first need to know the the taxonomy slug which we can simply provide in a 1 instance that would return all the terms of that current post, but when it comes to the child terms we will need to specify the parent's one by one in order to get the children of that current taxonomy term.
We do not have a shortcode to return only one parent term, instead our shortcode returns all the terms that have been attached to the post.
So the simplest way of doing this is using the nested view as it is able to do exactly what you want 🙂 just that it may look over complicated.
I will be more than happy to help you with this setup if you like 🙂