Add a link to a tableselect row
I solved my issue.
The links column was not rendered for the tableselect form because I didn't update my $header array with the new element name edit_link.
It appears a table form does not map the element names to the header names, meaning I did not have to update the header array for my link columns to appear.  As long as the element name was in the header array Tableselect would render the column.  I believe this to be the basis of my issue.