It is nothing. For public housing to make sense it has to be a considerable amount of the supply, and be universally available, not only for low-income people.
[1]> NYCHA has approximately 13,000 employees serving about 173,946 families and approximately 392,259 authorized residents. Based on the 2010 census, NYCHA's Public Housing represents 8.2% of the city's rental apartments and is home to 4.9% of the city's population. NYCHA residents and Section 8 voucher holders combined occupy 12.4% of the city's rental apartments.
1 out of 20 of the city's residents, 1 in 8 if you count voucher programs. That's far from nothing, but you're right, there's definitely room for improvement if you can cut through the red tape of nimbyism and corruption.
It’s not ideal.