I don't see the argument in what you are saying.
Housing prices go up. Developers move in and scoop up desirable properties, remodel the kitchen, paint it grey, and sell to a young tech worker for 3-4x what they bought the property for or demo a 100 year old beautiful greystone and put up a modern shipping container on stilts with 8 apartments and no parking.
Property values go up, long time residents are getting older. The ones who CAN move sell first and you get a runaway housing bubble. The ones who CAN'T move? They're on a fixed income. They need public services but they can't afford more taxes. Property taxes go up. Sometimes older people on fixed income can get a freeze, but taxes go up anyway. More people, more services.
People don't like taxes going up so either you put it off as long as you can while services atrophy and you make it the next guy's problem, or eventually schools start going bankrupt. Teachers go on strike, not enough police, not enough money to keep up transit and infrastructure, and you've got a lot of upper middle class people living in a very expensive post-apocalyptic hell hole.
That's Chicago.
People see and fear gentrification. Granted, it can't be stopped, but it shouldn't be ignored.
First of all the prices have gone up long before Amazon came in that's just how New York work them moving in will affect housing prices NOT renting prices.
There is no way you can spin this in favor of the local politicians. They just cost many of their fellow neighbours 25K jobs many of them high paying which is important in New York.
If such people truly have enough power to quash the NYC deal (I'm not convinced they do), then isn't it worthwhile to figure out how to allay their fears? Do they have a shot at the new jobs? Do they have the necessary skills? Is training being offered? Will their schools become overcrowded? Is rent control adequate? Will their unit be upscaled or bulldozed?
Most humans are adverse to change, so the rate of change is also a factor.
It's really that simple. You don't need to turn them into victims or anything like that to explain it.
There is nothing new about the rate of change with regards to Amazon. Had it been another company you wouldn't even have heard about it and it would have been installed. It's purely political opportunism.
All that is left is that these politicians cost new york 25K jobs. That's it.