They only say "license and registration" in NY. They don't need to ask for proof of insurance. That's all in the computer system. License allows them to check you, the driver. Registration allows them to check the car, and whether or not it's insured. The moment your insurance lapses in NY, the insurance company is required by law to notify the state DMV. I'm not sure, but I wouldn't be surprised if they have the information on your inspection as well (i.e., you can't just get an inspection sticker on some black market, because the vehicle must be hooked up to a computer that connects to the state DMV for your car to pass, unless it's older than a certain year.)
So, that's the sensible way to do it. If you let your insurance lapse, you'll get mail immediately from the state DMV that says, "surrender your plates." A simple run of your plate number will let any officer know if your insurance is current. And, despite insurance being god awful in NY, it's also why we have one of the lowest percentages of uninsured drivers in the US. (NY is the 3rd lowest.) It's really nice to know that someone didn't just buy a $500 beater, get to keep the plates from the previous owner (that's a really stupid policy), and if they get into an accident, have no insurance, and no way to pay for the damages they did.
Egads, there are states with more than 20% uninsured drivers on the road. It appears that the people in some states really ought to rethink how you're able to check for insurance and get those idiots off the road.
http://www.iii.org/fact-statistic/uninsured-motorists