FIDDLER is too dark for television, especially at Christmastime. And there's that whole Jewish thing. (It will have to come with Yiddish-to-Middle America subtitles. Do they know what 'mazel tov' means in Tennessee?)
HTSIBWRT is a great idea, except it's not family fare. And it's a musical satire. Without audience laughter, it's going to just lie there.
MY FAIR LADY is a very long show, about 2 3/4 hours, IIRC. They won't be able to shoehorn it into a 3-hour time slot and get the number of commercials they want.
BIRDIE may work, but the memory of the dreadful Jason Alexander/Vanessa Williams endeavor may kill it. Besides, those who were teenagers in 1960 are now grandparents (or great-grandparents if you live in Arkansas) so I don't see it having much mass appeal for today's families.
I vote for OLIVER!
'Our whole family shouts. It comes from us livin' so close to the railroad tracks'