How to display taxonomy terms the way they are hierarchical

term_group seemed to have never been fully developed together with the alias_of parameter. What the exact indended use is (or was), I cannot say for sure. I would just rather avoid the use of those two parameters.

Regarding your issue of creating a hierarchical tree, wp_list_categories() immediately jumps to mind (remember this works for custom taxonomies as well). You can either use wp_list_categories() as-is and manipulate the output through the parameters availble, or you can look at walk_category_tree() which wp_list_categories() uses to build its tree. walk_category_tree() also use the Walker_Category class, so you can use this to exact fine tune what you need

There is also the option of writing your own recursive function to create your tree

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