State Farm does the same the same thing. You call and at the time they will offer vendors to replace your glass and even set up an appointment at your home. The process is maybe 10 minutes. They pay the shop directly. It's a vendor that they use when you make a glass claim. GEICO may even use the same vendor.I have Geico as well, and yes it is normally part of their comprehensive coverage in their auto insurance plans.
So the only variable is what deductible did you select for comprehensive coverage.
In addition, unlike some insurance companies that send the owner off to get the windshield service provider and replacement on their own, and then reimburse the owner... Geico handles everything and generally uses Safelite with aftermarket glass. They set up the appointment (at your home, or in the shop, your choice) and they handle all the price negotiations and payment directly. You simply let Safelite have access to your vehicle. The only rub is that if you need recalibration of the camera.. that generally needs to be done at a dealer (most Safelite shops are not equipped for it). I discussed this in advance with Geico, due to some reports of Honda dealers refusing to calibrate a non-OEM windshield. Geico said that if calibration was needed, they would pay for it... AND if the dealer refused to do so with aftermarket glass.. Geico would pay to have it replaced with OEM glass. I got it in writing from Geico before having the glass replaced.. but never needed to recalibrate, because a careful road test after the glass was replaced showed all camera related HondaSensing to be working the same as pre-glass change.
In my experience Geico is extremely customer focused on insurance claims and repairs, and they have never let me down in more than 20 years, nor stuck their thumb in my eye over a claim.
If you read the posts some people do not have glass coverage which may be due to the customer's choice, state law or not offered by the insurance company. My point is the statement that it is "free in NY" is not true.