I think it has to do with ambiance.
In going to a Chinese restaurant, I expect to see Chinese people.
If I go to an Italian restaurant, there should be Italians.
If I go to an American restaurant, I expect to see fat people.
It’s worth noting the factor you are pointing to. The people the OP identified as “Asians” may be from a wide range of people.
Staff in “Indian” restaurants in Dublin often tell me the operations are owned and staffed by Bangladeshis or Pakistanis. Few seem to be operated by Indians. It’s why a small number of “Indian” restaurants don’t supply alcohol, although they will open a bottle for you. The owners are Muslim.
As people pointed out above, it’s about branding. Customers know the food range implied by the Indian brand, and few can distinguish between peoples from the sub-continent.
Although you’re correct that there’s an ersatz globalised Irish pub system, there’s also, especially in Britain and North America, a long lineage of independent Irish-owned and run, neighbourhood bars and pubs. The bizarre thing is, the phenomenon of the fake Irish pub exists here too.
There’s a Chinese restaurant in our town where ALL the staff is from Mexico except the Owner/Hostess.
This is where I read about the insular American-Chinese food economy:
The Fortune Cookie Chronicles: Adventures in the World of Chinese Food / Jennifer 8. Lee.
There’s a small region in China where very many immigrants working in the Chinese food industry come from, and once they are in a major city like NYC can find work advertised all around the country.
I was going to post a WAG that their margins were so tight, given the costs vs. earnings, that they had to only hire within their family, and any outsiders were probably working for less money, possibly to gain experience so they could start the same sort of business.
Since you know first hand, can you confirm my guess? Is the costs of supplies, and upkeep so high relative to earnings that most of the workers are unpaid children, co-owner spouse, and pseudo-journeyman cook/helper?
Almost all of the ones around here (San Jose, CA) have many Hispanics working in the kitchen and as busboys. I’ve even had Hispanic waiters occasionally; I presume those ones got promoted from busboy. Otherwise it’s a family thing and probably some self-selecting among job applicants.
There is also very little dairy (crab ragoon is all that comes to mind) and it’s easy to avoid the pork and shrimp. 
I visited a friend in Missoula, Montana many years ago and we stopped in at a Chinese takeout place, staffed entirely with white college kids. I was extremely amused.
Student visas only allow students to work on campus: if the Tim Hortons my have fit the criteria.
I once went with my uncle Carlo, who’s from the old country, to one of these tiny restaurants in Little Italy. We were a little surprised to see that the waiter was obviously Chinese, and that he spoke to us in flawless Italian. Through the whole meal – greeting us, taking our order, asking about the food.
My uncle was sufficiently impressed that he called the manager over afterward and asked him, “Where did you find this waiter? He speaks Italian so well.”
The manager got really flustered and hissed, “Shhhhh! He thinks we’re teaching him English!”
In my experience, most Indian restaurants serve Punjabi food, so it’s not surprising to me that the staff would be Pakistani.
We went to a Japanese restaurant in North Carolina. It was a steakhouse place with the cooking table deal. We were the only Asian customers in the entire place. We are Chinese, but I doubt the restaurant could tell the difference. As far as we could tell there was exactly one half-asian worker in the front of the house. And of course they assigned him to be our waiter.
Lots of Irish pubs on this side of the world, including several in Bangkok and upcountry. One, The Dubliner (with a branch in Singapore), actually is Irish-owned. For years, it was run by an Irishwoman, but she was eventually arrested for being part of a large boiler-room scam, along with a few other Irish and scads of other nationalities.
Dare I ask what a “boiler room scam” is? Are, er, “lady boys” involved? Elephants? 
Oh, the Metal Building salesmen I worked for!
