Why don't aircraft have DME that can automatically convert the slant range to ground range?
There's no accurate and simple way to know the height above the station in order to do the calculation with the existing equipment. Altitude is not an indicator as the DME station may be above sea level. If your airplane is at 15,000ft and the station is at sea level you'll get one answer, if the station is at 10,000ft you'd get a completely different result, and mistakes could compromise safety. 

The only way your proposed system would work is if the DME device had a complete database of every station height and altimeter readings, which would require upgrading the DME devices at great expense. I don't think anyone is interested in that when you can have GPS for a lot less money.