There are quite a few reasons. First off, as everyone has already mentioned, HWAs are quite fragile and somewhat more complicated to decode (though the latter is a non-issue with modern electronics). More importantly though, a pitot-static measurement offers some distinct advantages specifically as far as aviation is concerned.
On the face of it, a pitot-static system is very simple, both in concept and implementation. A typical integrated pitot-static tube measures total pressure at the pitot port and has a series of equally spaced holes around its circumference (in the cylindrical portion) to measure static pressure. In common use are also pure pitot tubes, wherein a static reference comes from the static port altimeters are connected to.
It is important to note that the tube itself does no measurement. Merely, it provides a reference location for a sensing element to be connected via pipework of some type (usually a mixture of rigid and flexible hose, depending on the application). It is at the sensing element where the magic happens.
When a differential manometer is connected across the total and static references, it is able to directly measure dynamic pressure in that location. This follows from incompressible flow and the Bernoulli equation, namely $P = p + q$, where $P$ is total pressure, $p$ is static pressure and $q$ is dynamic pressure. Upon expansion, $q = 1/2 \rho U_{\mathrm{inf}}^{2}$ where $U_\mathrm{inf}$ is the free stream velocity of the airflow giving rise to $q$, and rho ($\rho$) is the free stream air density.
When we assert a constant rho of 1.225 $kg/m^3$, we can rearrange for free stream velocity: $U_\mathrm{inf} = \sqrt{2 q / \rho}$, or if we factor out $k = \sqrt{2 / \rho}$, $U_\mathrm{inf} = k \sqrt{q}$ - a simple square root relationship which can be implemented mechanically or in software quite easily.
Where this is particularly useful is in aerodynamic calculations during flight. As every important flight parameter (lift, drag, control input, etc.) is directly correlated to $q$, having an indicated airspeed (IAS) also directly correlated to q simplifies a pilot's job drastically. There is now ONE stall speed for each configuration. The feel of the aircraft is exactly the same, no matter the air temperature, the list goes on. The value of this simplification in reducing pilot workload cannot be overstated.
This is why, for any normal flight operations outside navigation, indicated airspeed derived from directly measured dynamic pressure is by far the most useful speed measurement to have. Navigation is secondary and for that there are conversion tables, DME and GPS.
It should be noted, that the raw $q$ measurement from the pitot-static system is almost never left unadjusted for commercial and military applications. Anything from the pitot port's position to compressibility effects must be adjusted for in order to preserve the IAS relationship a pilot expects, which is why modern commercial and military aircraft first convert IAS to calibrated airspeed (CAS), which accounts for the pitot-static system's position error and then into equivalent airspeed (EAS), which further accounts for compressibility as soon as the incompressible flow assumption breaks down (usually at any Mach number above 0.3).
It should further be noted that both Reynolds number (related to true airspeed or TAS) and Mach number will have an effect on aerodynamic forces, but those effects are usually small (during subsonic and low transonic flight) compared to the effect of dynamic pressure and may or may not be accounted for in other ways (i.e. adjustments to the fly by wire system, if present).