Please do time the lights if you think there's a problem. I certainly can't claim that every single yellow phase in the city lasts a full three seconds.
I realize this is the case more in Manhattan, where I live, than in the other boroughs, but on streets with progressively timed signals, slamming in the brakes is rarely necessary, since everything is quite predictable. (Unless I'm coming to a major cross street or a strange intersection, if I've been hitting greens while driving at the speed limit, I will continue to hit greens.)
The yellow light is currently a message that those who can stop before the intersection should do so, or they run the serious risk of passing a red light. Extend the yellow phase and that message is a lot less direct. If the yellow lasts 5 or 7 or 10 seconds, drivers approaching it won't have any idea when it will turn red. Contrary to what you suggest, extending the yellow won't stop the impatient from running the red, as they already do, but will make it harder to avoid running the red for those who care.
I believe the cameras will cite those who run the red at any point during its phase -- except, IINM, the first half-second or so. Yes, there is a grace period. IMO, the yellow phase is already a grace period; tacking on an additional implicit grace period is a mistake. (When it's yellow, it's okay; when it's green, it isn't. It's a lot easier to convince people to follow the rules when they're clear-cut like that. Perhaps we could extend your point of view to conclude that it's okay to hit a fraction of a pedestrian!)
As long as a vehicle has entered the intersection before the light turns red, it's in the clear. The cameras only nab vehicles that enter the intersection on red. It's not a problem if the light turns red while you're still halfway across Queens Boulevard. If the yellow were longer, cars would continue to (legally) enter the intersection during the extended yellow phase.
Zero-tolerance towards red light running would force a lot of drivers to change their attitudes. Aggressive driving is too amorphous a category to enforce. I'd like to see the NYPD to increase its enforcement against those who run red lights, fail to yield to pedestrians while turning and where otherwise required by law, fail to signal their turns and lane changes, stop or park in travel lanes on multilane streets, block bus stops, and drive or park in bus lanes. (Let progressive timing take care of the speeders. Nobody around here bothers speeding for more than a block or two because there's no point.)