I would imagine these kinds of Chinese Restaurants would be more analogous in numbers and styles to the local chip and curry shops in GB.
These kinds of carry-out chinese restaurants I am talking about are also in contrast to the more established and classical full-service sit-down chinese restaurants of my youth that usually operate as a full service restaurant and usually have a higher caliber of cuisine. The type of restaurant that I assume the OP is talking about is more of a hybrid chinese fast food operation usually with little or no sit down service.
You don’t necessarily have to look towards the more upscale or established places for specialty dishes.
小吃 “xiao chi” - literally “small eats” - is very popular. In Asia, they are often found in small, unassuming storefronts, night markets, stands, holes in the alley, etc, and they consist of food that is popular in that region. For example, Tainan xiao chi is pretty famous in Taiwan. There’s Beijing xiao chi, Shanghai xiao chi, etc etc.
Xiao chi in America is mostly found in less unassuming places (as in, not a stall), but they remain low key, quick, and not known for good service. Hole in the wall vibe. Definitely not upscale. Not a bad choice for takeout. Tables are often shared, and the establishments are generally packed with Asian clientele, generating a huge waiting list… even if the food is not as good as the real deal. They are not really as specialized, as they also carry dishes that are popular from other regions. A place claiming to have Tainan xiao chi may carry a few special dishes, but also stuff that are popular in most of Taiwan in general and even popular things from China. For ex…
I looked up one of the Tainan xiao chi places from my hometown (in California), and I was mildly surprised to see that their web page was all in Chinese… no English at all. Here are some screen shots from the page for some examples of specialty dishes. Dan zai mian is a dish that Tainan xiao chi is known for. Tofu kan, tong zi mi gao (savory, glutinous rice) and taro & porkchop soup are more like general dishes. And xiao long bao is from Shanghai; I suppose they also carry dishes that have universal appeal amongst their Asian clientele (who doesn’t like xiao long bao?!)
I assume the type of restaurant that the OP is talking about is also almost ubiquitously characterized by their Lunch and Dinner combination plates which usually consist of a choice of egg drop or wonton soup, an egg roll, and a plate of pork fried rice and a choice of 30 or so chicken, shrimp, pork and beef classics- lo mein, General’s chicken, Black Bean, Sweet and Sour, Szechuan, Kung Pao, etc. all between 5$ and 7$.