INS cannot give you airspeed and GPS does not help
GPS works really well in combination with an inertial navigation system to give your current position, speed etc. All of this ends up being rerenced to an earth based coordinate system via the GPS. This is very useful but not if you want the velocity of the aircraft relative to the air surrounding it. In this case you either need an independant measure of the air velocity relative to the ground and this in general seems totally impractical or you need to measure the speed with respect to the aircraft which brings you back to where you started. An INS is no use at all for measuring airspeed. It is perfect for measuring ground speed.
On why they do not use doppler to measure airspeed I am not an engineer working in this area but it may well be because you need to bounce a signal of a known frequency and velocity off something moving wth the air. I imagine there is not always anything there to bounce off. There are lots of ways of measuring gas velocity but I am sure the possible methods, their reliability, operating ranges, failure modes etc have been analysed extensively over the last 7 decades or so and what is used is used for good reasons.