Jason Kuznicki asks why do they do it? Why don't they make the "secret menu" common knowledge? He gives some answers, including:
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.
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.
Thai restaurants don't have secret menus per se, but often you can talk a so-so restaurant into, for your sake, becoming a very good restaurant with real Thai food.