Check out a Specialized Turbo Vado SL. The biggest negative is price, but it's in your budget. I have the 5.0 EQ model and love it. Very natural feel to it with assist that's seamless but definitely doubles your output. It's also probably the lightest commuter-style ebike you'll find. NYC roads aren't any worse that Houston's, where I ride. I put a RedShift suspension seatpost and Brooks B17 saddle and it's like a magic carpet. Even on maximum assist and riding flat-out I can get 30+ miles on a charge. Dial that back to Eco mode with 35% assist and it's nearly triple that.
https://bikerumor.com/2021/02/28/rev...and-urbanites/