I cycle daily in NYC and have used both Citymapper and Google Maps

Google Maps' cycling instructions route me onto streets with bicycle lanes. I haven't had any problems with it and I think it's a fine option. Maybe because NYC is the type of location google has excellent data on.

Citymapper is also good. The fastest vs quietest distinction doesn't often yield much of a change in route. What I like about citymapper is you can distinguish between riding a personal bike, or riding a bike rental. Sometimes when bringing a friend along, they rent a citibike, and it's nice to have the option for directions to take you to a docking station close to the destination.