I'd like to just discuss the word 'entitlement' briefly because I wonder if the emotion behind this word is making the situation seem to be something it's not. The thoughts about stage doors prompts me to make this comment. Basically, I do not think people are 'entitled' to see a performer in the sense that if they pay $100 it's their god-given right to see someone perform in front of them. I especially do not think that people are entitled to see a performer at a stage door (I personally rarely stage door, and it is incredibly annoying to hear people harp on about how an actor is particularly annoying because they never stage door).
I suppose what I really think they are 'entitled' to is a refund or exchange for a performance that was not as advertised. Now, I will explore this a little further by responding to this comment:
"And with extremely limited exceptions no producer ever guarantees anything other than that the show will be performed approximately at the time listed on the ticket. Because they can't. Let's get this straight: you are not entitled to a damn thing other than that."
Now, I acknowledge that 'legally' the obligation is for the show and not to see a specific performer. However, I really think there is a large difference between what is 'legally promised' and what actually exists in reality in terms of advertising and marketing a show. Yes, a show might not "guarantee" that the star will be in a show because as you said, they can't, but for all intents and purposes in terms of the marketing the show they are suggesting that if we buy tickets to a certain show we will see a certain performer in the show. Obviously many producers (particularly when the 'name is above the title') acknowledge this and therefore do refund/exchange if a performer is out. But I wish this were a more widespread practice. And what is particularly frustrating is that shows will very clearly advertise a performer in their show when it is convenient for them (i.e., convincing someone to buy a ticket), but if they can't actually deliver on these goods they are not willing to compromise.
A great example recently was Pippin. TONY AWARD WINNER PATINA MILLER occasionally missed shows, but (as far as my understanding goes) they weren't willing to refund or exchange tickets. I think it's really a questionable marketing practice that they can get away with convincing people to buy tickets to see a Tony Award winner star in a show and not refund/exchange when they can't deliver the goods. Of course, because it is such a widespread and apparently legal practice they can get away with it.
I'm happy to have the argument another time about what the experience of seeing an understudy is like (yes, I am very happy I caught Jeremy Jordan in West Side Story but no, I was not happy I missed Patina Miller) - but that is not the core issue here. The core issue for me is that shows are marketing/advertising that they are providing a certain service, and when they can't they aren't obligated to be accountable for that. That seems wrong to me.
I know the counter-argument will be "but they aren't promising they are providing that cast, only the show". And yes, maybe 'legally' - but as I said for all practical purposes and how shows are sold in the real world, they are suggesting you will indeed see a certain performer if you see the show.
I'm looking at Something Rotten!'s website and it's just a classic example. Assuming they don't give refunds/exchanges, look how they are advertising the show on the front page:

If they are able to advertise the show and suggest that there will be a Tony Award winning performer in the show, I really think they should also be obligated to be held accountable (in terms of refunds/exchanges) if they can't deliver on what they advertised.
Give me claws and a hunch, just away from this bunch.