I don't think it's private, from their website it sounds like individual permission from owners/operators is not required to visit. Destination wouldn't be wrong but I also think it's implied by the fact that the roads have no exit. 

We could tag the car gate opening hours per https://www.catholic-cemeteries.ca/gate-closing-times/

However the wider problem of snapping to an inoptimal access road is a longstanding problem with most OSM tooling and as far as I know there isn't a solution so far. Maybe mapping the wall visible on satellite imagery would help, if it doesn't I'm not sure there's much we can accurately do.