I've only found it worth my while to get the menu for Koreans on rare occasions and I don't know what I'd even want off an authentic Chinese menu. But sometimes it's fun to peruse the ethnic menu.
Anyway, a couple of economics bloggers I read tried to sum up the economics of keeping the Chinese-only menu secret.
1. Path dependence (a): Americans have some very set though inaccurate ideas about what “Chinese food” really is. They will generally balk at anything else. More people will break this way, and avoid the restaurants, than will break my way, and go to them more often, if they are offered something new and different.
2. Path dependence (b): Setting up a restaurant is a ton of work. Someone or some entity tells Chinese restaurants what they must to sell to appeal to Americans, and all the restaurants are following the same bad advice. The agent(s) to blame aren’t as subject to market forces because Chinese immigrants have fewer contacts than most others in America. If this seems speculative, consider how few different brands of chopsticks you’ve seen at Chinese restaurants, from the fabulous ones to the truly wretched. There aren’t that many.
3. I hate to bring up the obvious, but… chauvinism. Chinese people have certain ideas about Americans, including that our culinary tastes are incredibly narrow. Obviously, this may be partially true, given (1) above.
4. The high costs of offering so many different dishes. I’m skeptical of this one, because Chinese people are usually offered the Chinese menu, if there is one, while Americans get the American menu. The costs of being able to prepare the dishes are in place either way.
Tyler Cowen offers this.
I would add that perhaps many Chinese restaurants do not want too many non-Chinese customers. Especially for immigrants, restaurant life is often about ambience, social contacts, and feeling you have a space to call your own. A restaurant cannot be all things to all people and the #1 best way of judging a restaurant is to look at its customers. The "beef with broccoli" menu will attract a certain kind of American customer, but without breaking down the sense of segregation and the basic Chineseness of the place.
That said, there is also the fear that the American customers will order from the secret menu and then not like the chicken feet, etc. and give a bad report to their friends.
I think they're both overthinking it. I would submit that the Chinese proprietor knows that Mary Homemaker has no interest in an authentic Chinese meal, they just want their beef and broccoli takeout.