Source for the labor existing without immigration? There are tens of millions of people retiring compared to turning eighteen. Increasing pay should increase labor force participation a bit. But it doesn't solve the fundamental demographic problem.
[1] https://www.businessinsider.com/millennials-living-at-home-a...
Plenty of people want to come to the US btw, if you look at all the people coming over the southern border illegally. We just need programs to help those people do it legally, distribute them across the country (instead of concentrated in border states) and help them get acclimated to american culture.
The comment argues labour shortages are a fiction used to drive down wages. Immigration solves a labour shortage by driving down (or stabilizing rising) wages.
If wages for other jobs increased a lot, servers could be enticed to leave the restaurant industry, and go elsewhere. No one really needs waiters. Customers can instead go to the counter and order themselves, and pick up their own food (this is usually called "counter service"). Eliminating all the servers in the US would free up a LOT of labor for more important tasks. This is just one example. People paid to pump gas at full-service gas stations is another (which has largely disappeared, except in two states, NJ and OR).
You're describing totally different modes of dining. That said, this is probably the direction we're heading: servers will make more. And dining out (actual dining out, not fast casual) will become more expensive.
That said, you have a point. We have a lot of make-work jobs. cough cough TSA. We also have a lot of jobs that look likely to go away to automation, e.g. long-distance trucking and certain warehouse roles. So maybe this balances out on its own without consumer prices rising too much.
I'm also describing how the high (and getting higher) cost of dining in America is partly a function of American expectations. In other countries, dining is much cheaper, but from what I see, there's far less labor involved. Here in Japan, eating out is pretty cheap usually. (It's similar in Europe.) But there's less service: you have to actually yell for a server sometimes, or otherwise get their attention, because they're busy serving other tables or doing other tasks. They don't have time to come check on you, and in fact it would be considered rude if they interrupted your conversation just to ask you how the food tastes, but this is perfectly normal in America.
If American restaurants adopted a different service model and standards, costs would be lower, it's pretty simple. No one really needs servers hovering over them and chit-chatting with them and acting like their best friend.
I don't like to use TSA as an example because it's a government service, which doesn't operate under market pressure the same way as private business. My point is there's a lot of unnecessary labor even in privately-run businesses.