If you allow me to comment on two of the common misconceptions mentioned earlier...
* GPS does not directly give you height above sea level. It gives you height above the WGS84
ellipsoid. By applying a correction based on a stored
geoid model, this can be translated to a mean sea level height, and most non-specialist GPS receivers have the ability to do that.
* While it is correct to say that the biggest uncertainty in a GPS position is in the height plane, such uncertainty is by no means worse than achievable by a barometric altimeter. IIRC, just the allowed instrument error of a primary reference altimeter is in the order of 100ft at altitude. You must add to that all the other sources of errors and your barometric height can easily be thousands of feet off, as you well know (which is how we all grew to hate the CRP-5). By comparison, an uncorrected single frequency GPS height (SA off) is usually not much worse than about 30m. Differential dual frequency GPS gets this down to usually not more than 0.5m 2-sigma (e.g.,
Starfire systems), and in the case of GPS the vertical error is independent of altitude.
I am not qualified to comment on the use of GPS in the field of aeronautics, but I would guess that the reason for your barometric altimeter being used as the primary altitude reference is not related to its precision or accuracy (of which it doesn't have much), but rather it owes more to its simplicity, to its working being independent from external factors other than having an atmosphere, and to its availability. At the end of the day, you could say it doesn't really matter how wrong your altitude is, as long as it's wrong by the same amount as everyone else's.
Hope this helps