Most cities in the world have areas which are more expensive and least expensive. Very few cities will be only for the rich (where do the restaurant employees live), and few big cities will have no rich people. There are of course small towns which have only poverty or only extremely rich people, but those are the exception.
So when a city looks at their underserved areas they will often want to provide the same quality of service to the underserved areas as they do to the wealthier areas. People ask why the poor areas cannot have access to parks, transit, walking paths, and other valuable services. This is a good question to ask.
The issue then becomes that by building these amenities the desirability of the lower-income neighborhood will increase and more people will want to move there. Increase demand, increase price.
So then a city might say that they want to improve everywhere a little bit all at the same time, which is an admirable goal, but in reality, the world can’t work that way. Cities have limited resources in terms of labor, construction machinery, and the like. I’m not even worrying about money yet. So some areas will necessarily get the amenity before others. If a city does this long enough to build a well-connected transit system, expansive trails, good quality parks, and other amenities that people enjoy, then the city will become a more valuable place to live compared to other cities in the same country.
For cities in the United States, the cities that have built the most transit and walkability are now the most expensive. Cities which have not invested in these amenities are less expensive. City-states can choose to limit who can and cannot live in their city through a visa policy, but the only city that is like this is Singapore. Monaco, Malta, and Vatican City are all part of the Schengen area. Singapore is also the third most expensive country in the world to live in, behind only the Cayman Islands and Switzerland. This does not adjust for wages. So even by controlling immigration you still have the reality that if you are a low-corruption city with great job opportunities and great urbanism people will want to move there.
The only way I know of that would make it so average people could live in great urbanist cities is if everywhere started building in this way all at the same time. This is impossible. So the best we can do is improve our places or relocate.