A ten to fifteen minute walk is less than a mile, unless heavy drinking somehow speeds one up. Where does that distance cost $20 in a cab? The NYC taxi & limousine website shows one mile in Manhattan as costing about $6.30--$5 for the fare, .80 for MTA and improvement surcharges, .50 for night surcharge.
NYC actually has far cheaper cabs than most US cities (who don't have Uber) because the market is extremely different. High demand and high supply 24/7 as opposed to surges. The pricing is also more tightly regulated iirc (haven't looked into the specifics).
When I took the train up to Albany, I wanted a cab to a coffee shop across the bridge. It was a half hour walk vs a 7min car ride. The cab was flat rate into the city so it would have cost $20+ just to take me across the bridge. This would've been a sub-$10 ride in NYC. I walked it, despite the rain.
I just saw a stat yesterday that NYC taxi medallions reached their lowest value with a total of 13.6k medallions in existence; compare with 50k Uber/Lyft drivers.
It's probably not just the cost but finding a cab at all. You have to call them and will likely go home when everyone leaves. Without surge pricing, there's no fluctuation in supply so that you don't get a cab.
That's my experience at least. It's also cheaper in the end but with an uber you know pretty well when you'll be in the car.
Edit: My mistake, misread the text.