I believe that most of the Tony performances are done to backing tracks that are recorded shortly before the telecast by the Tony orchestra, often with a few guest musicians from the show's pit orchestra. Using cast recordings for this purpose would create a nightmare of reuse fee payments, and the Tony performances are frequently ridiculous medleys that have little to do with the structure of the cast album songs (or common sense).
"I have got to have some professional music!" - Big Edie