I am a little cynical when it comes to their selections. The networks take virtually no risks in their selection...did we really need Peter Pan or Hairspray, for example? The show needs to be something that people are familiar with because it led to a big movie hit, e.g., the too-soon Hairspray, and / or one of the small list of shows that has entered into the zeitgeist, e.g., The Sound of Music, Peter Pan, Grease (because of the movie more than the original show which didn't have all the songs audiences sang along with), Rocky Horror (also because of the movie) and MAYBE A Christmas Story, but ONLY because of the movie on which it was based, which has entered into the zeitgeist. Of the shows I can think of, only the Wiz didn't seem to fit either qualification, although I know I am missing a couple.
That said, I actually think there are a limited number of options (the names suggested are mostly illustrations, not real suggestions, except #1):
1. Hello, Dolly with Bette Midler and DHP. Probably, her only chance to get it saved for posterity (for real audiences), and people who NEVER go to the theatre were aware of that production; hell, a number of those people saw it). 'Legend Bette Midler repeats her Tony Award winning performance...).
2. The King and I. Seems perfect, but I can't think of any star names to put into the show, which seems a requirement to attract audiences. There is no male Asian star that I can think of who would cause audiences to dial in. Also, I can't actually think of a female star who would be right for the role AND attract an audience, other than MAYBE Amy Adams. Sad to say, but the fact that the large percentage of the cast are Asian makes it difficult to impossible to sprinkle stars throughout the show. Were it not for those reasons, I think it'd be a perfect show to present.
3. Chicago. Much too soon IMO, but more time has elapsed since the Chicago movie than with Hairspray. It would clearly depend on the casting, e.g., Mylie Cyrus for Roxy Hart.
4. La Cage Aux Folles if they can get some really big stars for the lead. It has the advantage and the disadvantage of never having been a movie. How about Hugh Jackman in the role of Georges; how about (can't think of a name for Albin...Nathan Lane has already done it, and is probably too old at this point anyway).
5. Mame. It doesn't seem like a Broadway revival is ever going to come along. Anne Hathaway or Amy Adams as Mame or Lady Gaga if she is any good in her movie. Amy Shumer as Gooch. Catherine Zeta Jones as Vera. A teeny bopper for Adult Patrick. A star cameo for Beau. A cameo by Angela Lansbury as Mother Burns.
6. 1776. Here, you could focus on seeing the history. Sprinkle well-known people throughout the cast, so people would at least get some name recognition.
7. Guys and Dolls, but only with major names (or major breakthroughs) in the leading roles and well known people in a bunch of other roles,e.g., Jake G as Guy Masterson, Lady Gaga as Sara Brown, Kevin Hart as Nathan Detroit (if he can sing at all), the breakout whose name I can't remember from Girl's Trip as Miss Adelaide, if she can sing. Nathan Lane singing 'Sit Down, You're Rocking the Boat'. Otherwise, no one would watch it.
8. I don't see Oklahoma, Carousel, or Show Boat attracting large audiences. I don't see Sondheim attracting audiences. I originally thought the exception might be with one possible exception, A Funny Thing, but two issues: (1) who would they get who is famous and can stand up to the legendary performances of Zero Mostel, Phil Silvers and Nathan Lane; and (2) A Funny Thing works by actually being there...much of the score is either really mediocre IMO or 'precious', e.g., would TV audiences really enjoy 'I, Myles Gloriousus' (sic)? A lot of pruning would be required and a great comedy writer would be required to expand the comedy even further.