Probably a question of resources. NYC has about 60k migrants under their care right now (or so they claim), and they spent $5.5 billion in the current fiscal year (and estimated $12-15 billion over the next 3 years) to give them food, shelter, and medical care etc
Through July of this year, the USA as a whole has gotten about 820,500 new migrants who got notices to appear before an immigration judge (and that doesn’t account for migrants who never got caught or entered into the system).
So if NYC spent $5 billion to feed, shelter, and care for approximately 60k migrants, you’re looking at possibly $70 billion per year for the feds to do it for all of them, not even counting the costs of hiring more staff and building more shelters
That might be why they rely on the private sector, like this place:
https://ozanambrownsvillecenter.org/