Is a "GPS required" ILS (or other navaid-based) approach considered a "GPS approach" for alternate planning's sake when TSO-C129/196 gear is in use? If you have TSO-C129/TSO-C196 GPS gear (vs TSO-C145/146 WAAS/SBAS gear), you either have to have a non-GPS approach at your destination, or a non-GPS approach at your alternate.  If your choices are either conventional approaches or GPS-only approaches, this is simple to understand.

However, some approaches have arisen that are hybrid approaches -- combining GPS intermediate guidance with a ground based navaid (usually an ILS) for the final approach course guidance.  An example of this can be found at Jackson Hole, WY, where the KJAC ILS Z or LOC/DME RWY 19 approach is listed as "GPS REQUIRED" on the approach plate due to the use of GPS to set up a curved feeder leg onto the ILS final, bypassing some terrain to the north of the airport and allowing much lower minimums as a result.

Are these "hybrid" approaches considered the same as a GPS-only approach for alternate planning if you are TSO-C129/TSO-C196 equipped?  Or are they considered as conventional approaches for alternate planning purposes?
This is an excellent question.

However, since the approach you reference, KJAC ILS Z or LOC/DME Rwy 19, "requires" GPS, I would interpret it the same as a GPS-only approach for planning purposes and subject to the appropriate restrictions. Those are: either the destination or alternate, but not both can be a GPS-only approach airport for TSO-C129/TSO-C196 [GPS-no WAAS] equipped aircraft.  Both destination and alternate can be a GPS-only approach airport if the aircraft is equipped per TSO-C145/146 [WAAS].