Great report! Sorry to hear you had to backtrack numerous times. I should have shared more tracks with you.
not a big deal...just frustrating on google's side...
i remember looking at your track going into Luna Park and remembered after that you took a different route into the area...i think you cut north/south along the bottom of the hill which would have missed the private land...
we hit a gate here, 33.566036,-107.241407
and here, 33.592391,-107.235849
so we continued down rt 1 a bit more til here, 33.51256,-107.220953 which was a forest road that brought us up to the trails we needed...
heading toward chaco we had another shortcut off rt 9 that turned out to be private land, but rt 57 wasn't far down the road and we just took that into the canyon...