Is it possible to accurately measure airspeed on a bike, and is it useful?
There are devices designed for bicycles that measure airspeed. And older example is the iBike, but there are others. The iBike is intended to be used as a stand-alone power meter, and makes assumptions about the drag area and rolling resistance of the bike/rider, then combines those assumptions with speed and gradient information to estimate power. 

The other airspeed devices (for example, the Notio Konect) are used in conjunction with an on-bike power meter to determine aerodynamic drag; the on-bike power meter measures power, an on-bike speed sensor measures speed and acceleration, and (sometimes) other sensors will measure gradient and air density. From these measurements, aerodynamic and rolling resistance drag can be estimated. Of the airspeed measurement devices that are currently available as of January 2020, none measure yaw -- they only measure airspeed from directly ahead. There are a few devices that are in development that do measure yaw. 

This SE.bikes question and answer describes some of the calculations involved in the estimation. A brief description of how the iBike estimates power is discussed in this SE.bikes topic here.