..why does flying safely rely on something that is obviously liable to blackages or damage?
Further to the other answers already posted, you might wonder why something as crude as a pitot is still used. Well, the KISS principle is valid but what makes the pitot so spectacularly the right choice for the job is that it responds not just to the SPEED through the air but the density of the air. Consider a speedboat pitot (a common way to log the speed of a fast water craft). Going through water, by the time you get to ten knots or so there is quite a good pressure and that can indicate speed. Imagine watching that boat's speedo with the boat out of the water and a ten knot wind....obviously and intuitively you know that that it won't read anything like that speed. Similarly, in outer space it will read zero. What causes the 'lift' to keep the aircraft in the air depends not just on the airspeed but also the density of the air. The rarer the atmosphere the higher the airspeed needed to keep it flying. As we go higher, the atmospheric pressure drops (which is why we use a barometer to measure altitude) and the air becomes less dense, giving less lift. An understandable technical explanation can be seen here:
Aerospaceweb.org | Ask Us - Lift Equation
The pitot responds to speed and air density. A GPS gives you groundspeed, not speed through the air: when flying there is always wind - anywhere from the nose, side or tail, and the pitot looks forward, which is what we're really looking for as the shape of the surfaces are designed to provide lift with airspeed from the nose of the aircraft. An anemometer (cups) doesn't really respond to density. Gyros don't do it (already explained) and Inertial Navigation (INS) is really just a different way of providing what GPS now provides at lower cost and greater accuracy.
I once stuck the pitot back on my aeroplane with five-minute epoxy. The engineer had changed the tyres one day, and gone home. He'd obviously knocked the pitot as it was hanging by its tubing and no other engineer was around. Down the hardware store for some epoxy and I was rolling half an hour later. Dead simple technology. Note: Children, don't try this at home.