Aircraft DO transmit altitude while on the ground... look at the ADSB data that comes from the aircraft. If the data hasn't been edited, in the "alt" column you can see clearly altitude IS being transmitted even though the aircraft is on the ground and displaying GND status. The data can also include the corrected altitude and the pilot entered correction pressure that is supposed to yield the correct height above sea level. That resultant varies quite a bit also.
As to why there are altitude variations among aircraft simultaneously on the ground at the same time... varying typically about 200 feet, boggles the mind. It's hard to imagine that a digitally monitored analogue device that is supposed to be calibrated (once a year?), could get so out of whack. But, there's another contributing factor.
Older and less expensive transponders output altitude data in 100 foot increments (newer are 25 ft and a few at 10 ft resolution); this means that when the first pulse of 100 ft altitude appears, you are probably only about 51 feet when the transponder transmits the message. This step resolution transmission of altitude is usually smoothed by some method of averaging. But what do you do when an aircraft has the right altimeter setting but displays -100 feet when you're at 16 feet above sea level? And then you move to another aircraft on the ground 16 ft above sea level and it too has the correct altimeter setting but it reads 50 feet?
For municipal and County owned airports, they may opt to buy a service like WebTrak from Envirosuite so aircraft can be accurately tracked locally. But, in our area, Envirosuite/WebTrak solves the above problem by wiping out all ground data and substituting one of two values that correspond with the elevation of the airport (Geodetic or GPS). But one sin leads to another.
This clean up means that altitude data has to be 'adjusted' all the way up to about 800 feet. This process erases all the altitude differences of aircraft on the ground. I'm currently looking for data on normal variations of altimeters. I guess we could blame it on Cro-Magnon technology of pressure cans moving a spring, but what are the alternatives? Radar altimeters are very expensive. GPS has satellite time delays and depends on ellipsoids and Geoids, none of which are universally accepted. That still leaves us without a satisfying explanation.