Final Director: "ABC123 callsign only to tower XXX.XXX"
Initial Call to Tower:
"XXXX Tower, ABC123X a Whizzjet Max-9000 series with information Juliet with you established on the LLZ shortly descending on the glide 9.3 miles final runway 26L"
DEF456 goes around
Tower
tersely: "ABC123X continue"
When a busy runway is well utilised there can be just a few seconds windows for some calls to be made - miss your window, miss your opportunity. Callsign only* really can help controllers to hit that window of opportunity when required. If they want more information than you've given they can easily ask at a more appropriate time.
*even smaller units don't appreciate a life-story when they would prefer to be giving the urgent avoiding action or missed approach instructions