Taxonomy archive template to have conditional logic for displaying child categories

You are so very close here, you can almost smell it. Here is how you would complete your code:

You already have all your child terms stored in an array called $children. To display these, you simply need a foreach loop. Something like this will do the trick. (You can just extend on this, simply do a var_dump or print_r to get the available objects you can use)

foreach ( $children as $child ) {
    echo $child->name; //very basic, display term name
}

If you need to get a link to the term, make use get_term_link