There is a substantial distance between actual GPS/SBAS performance AND theoretical error modelling done to bound SBAS performance.
In the real world, HPL/VPL (horizontal/vertical protection levels) are in the 15 meter levels while measured errors are in the 2-3 meters level, rarely getting to 10 meters.
Actual errors are always 5-10 meters better than HPL/LPV levels.
The whole thing was modelled using theoretical worst case calculations that might never ever happen in the real world.
Essentially the FAA/EASA is just being way over zealous in what performance they allow LPV to be used for. LPV200 is only allowed to be used up to minimums, even in visual conditions, while ILS is allowed to be used for autoland below minimums if visual conditions are present.
Eventually FAA/EASA should allow for coupled LPV autopilot at least until 100 radar altitude.
Plus for now we're only using single frequency, GPS network only SBAS receivers. The next generation of SBAS will feature GPS+Galileo+Glonass+Compass+regional augmentation with dual frequency processing at the end user (aereal) receivers.
This will yield:
1 - Global LPV200 performance
2 - Currently we have a few minutes a day where LPV200 performance isn't available in the California coast and hours a day where LPV200 performance isn't available in northern Canada and Western Alaska. That should pretty much be gone. There could still be a few weak spots worldwide where LPV200 could be unavailable for a few minutes a day
3 - The ACTUAL performance of dual frequency/multi constellation SBAS should be good enough for CAT II approaches in the real world, but since mathematical models are used instead of real world performance, we might never get SBAS CAT II or it could take many decades.
The keys are:
With dual frequency service iono corrections are now calculated in real time at the end receiver.
With 4 constellations+regional we go from normal 9 GPS ranging sources to over 20 and sometimes 30 ranging sources. Actual typical accuracy should sub meter 99.99% of the time or CAT IIIc performance. But this is meaningless for aviation safety authorities, what matters to them is their ultra super duper pessimistic mathematical models that still predict errors that don't even allow for CAT II !
In conclusion, LPV200 is a precision approach. Even LPV250 is a precision approach too. That's in practice. But FAA/EASA don't like it.
There's also a conflict of interest in GBAS vs SBAS features. GBAS requires a per metro area installation and barely provides for full CAT I approaches today (including coupled landings) and eventually will provide CAT IIIa approaches. If SBAS could provide CAT IIIa, then GBAS is dead, and there are billions invested on that ! So people don't want SBAS features to invade their precious future GBAS performance envelope !