A lot of low income people in NYC are immigrants; for many groups of immigrants, NYC is one of if not the largest concentration of their ethnic group in the country. Being around people from your homeland gives you familiarity and comfort, access to cultural items that may not be readily available elsewhere, and a social support network. One example of this is that New York's Chinatown has developed a completely independent farm network to grow Asian vegetables that are not available normally: https://www.jstor.org/stable/10.7591/j.ctt20d89sd

An immigrant leaving New York for a cheaper COL metro is leaving all this behind for the great unknown.

One other important factor is that in New York City it is very easy to live without a car. For poor people, a car is easily an expense just as large as housing. New York City has housing cost of $1.6K a month but transportation cost of $6k a year [0]; St Louis has housing cost of $960 a month but transportation cost of $11k a year.[1]

[0] https://htaindex.cnt.org/fact-sheets/#?focus=place&gid=16861 [1]https://htaindex.cnt.org/fact-sheets/?focus=place&gid=14098