https://www.nycedc.com/program/relocation-and-employment-ass...
https://www.nycedc.com/program/industrial-commercial-abateme...
These programs are designed to encourage specific behavior, like companies moving to and making capital investments in particular areas of the city. Any company that does so qualified for a formulaic tax break. NYC did not offer any tax breaks to Amazon beyond its existing incentive programs.
It does appear to be a "don't hate the player, hate the game" scenario which I do think is worth looking at. I would love to see real discussion with details like the link you presented from both side, and not these emotionally charged garbage opinion pieces presented almost as articles with no sources or credibility.