All of the arguments against airing TV versions while the live show was still running on Broadway used to be applied to the idea of making the film version while the show was still running. It was once standard to add a clause when selling movie rights that the film couldn't be made until the Broadway production closed.
But as others have pointed out, films of PHANTOM, LES MIZ and IIRC FIDDLER have shown that at today's prices, audiences want to be sure they will like the show before they shell out hundreds of dollars. So a film (or in theory a TV production) actually helps the Broadway b.o. Shows like HELLO, DOLLY! and SOUND OF MUSIC continued to be revived on Broadway and touring the country based, in large part, on the public's familiarity with the film version.
A bigger issue, I think, is that broadcasting a live musical play on TV is not a matter of just setting up some cameras and filming the show in progress. (Yes, I know they do it with opera, but few operas are as physically busy as most musicals.) Nor is it a matter of merely assembling the cast on their day off, as is done with original cast vocal recordings.
To actually do the work necessary to produce a commercially viable video would be next to impossible while the same cast is performing live eight shows per week. (Yes, we all liked SOUTH PACIFIC and LIGHT IN THE PIAZZA on PBS, but we are not the general audience. And neither of those shows are exactly movement heavy.)
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For the record, I am NOT among those who thought GREASE on TV was an unqualified success. In fact, in the threads immediately after the production, only Hough and Hudgins got universally good responses. To me, the producers sacrificed every bit of drama and most of the comedy in their attempt to reproduce the film on sound stages. The actors were so busy racing from mark to mark that (with the above exceptions) they all seemed like middle-schoolers reciting lines by rote in their first time on stage.