For all I know maybe GPS's nowadays include a worldwide map that gives the altitude difference between spheroid and sea level, wherever you are. If so, they could give a good approximation to height above sea level. I'm just saying such an addition to WGS84 (based on thousands? millions? of additional measurements) would not be WGS84 itself.
Or put it this way-- once you've determined sea level, it's the same sea level whether your GPS is set to WGS84 or NAD27 or OSGB36. The WGS84 spheroid won't be at the same height as the NAD27 or the OSGB36, so the map of altitude corrections will be different for each-- but they're all aiming at the same result.