As to their new works, the only ones that seem to matter to the Balanchine faithful are those that come from in-house. Robbins, Peck and Wheeldon are or were all members of the Church of Balanchine.
One crucial choreographer you left out who belongs to a different nexus: Ratmansky. NYCB works with him regularly. In the past, they even tried to hire him as their resident choreographer. Many of his admirers are also what you call "Balanchine faithful" though I'm sure not all...
The Balanchine faithful more generally? Well I know what you are talking about, but the leadership of the company has not been as insular as your comments would suggest when it comes to commissions and productions. They commissioned two works from William Forsythe and did so considerably before other American companies were regularly performaning Forsythe (has ABT yet danced Forsythe at all?)...They also regularly perform & have at least 2 or 3 times commissioned from Bigonzetti and from Preljocaj--European choreographers well out of the Balanchine nexus and whose works are maintained in the repertory. They also commissioned a work from Ulysses Dove that for a while at least was being regularly revived--they could hardly commission additional works after his death. I will be silent on the one commission from Eifman since I doubt that experiment will be repeated anytime soon. But at any rate NYCB did do the experiment.
Maybe these choreographers are not as admired among some NYCB fans/critics as others, but that can't be held against the company's leadership or director who brought them in and keeps their works in the repertory...
Also at one point NYCB commissioned several works from Tharp, though I believe she is generally admired among those who admire Balanchine/Cunningham.
(Puzzlingly, I remember you asking on another thread why NYCB even bothers with choreographers other than Balanchine/Robbins when they are so great--perhaps you were being ironic and I missed it. That sometimes happens on the internet...)
The company's post-Balanchine turn to increasing the full-length story-ballets in the rep including Swan Lake. Romeo and Juliet, and now the 19th-century Bournonville La Sylphide is certainly not something I think we would have seen if Balanchine were still around.
The milking of personal connections to Balanchine that you dislike seems to me to work in good as well as bad or irritating ways. Companies that lose touch with their heritage/traditions will lose touch with what makes them great and distinctive. I think the Royal Ballet would have done well to cherish its Ashton connections more continuously in past decades. (Don't have a good sense of what is happening there now with Ashton.) And, for example, I don't want NYCB to start dancing Macmillan's Manon.
(Much of the greatness of the great Russian companies comes from the steady continuity of training and coaching from one generation to the next. NYCB has actually been strongly criticized by some of the "Balanchine faithful" for not doing enough with dancers of past generations.)
When a ballet company starts to try to be everything to everyone it ceases to be anything important at all--at the worst, it ceases to be a ballet company at all and morphs into a contemporary dance company. I am delighted that there are other companies with other aesthetics--that there is more to dance than ballet, and more to ballet than New York City Ballet. But, as you can tell, I love New York City Ballet.
That New York City Ballet has a style, an approach, a tradition that go back to its founders--and that it (re)presents those with reverence and sometimes arrogance? I think that's true. I also value what's great about that approach in an institution of such enormous quality and creativity. Above criticism? Certainly not. More insular than the Paris Opera Ballet? or the Mariinsky? Or the Royal Ballet? As institutions--I strongly doubt it.