Just as an update to this for anyone who cares, in taking out the remains of the shelving I discovered that the position of the crossbeams is adjustable, and they could've been shifted over and screwed into the wall studs. Instead they used more split-style drywall anchors.

That said, it looks like there were some parts screwed into the wall studs: three small hooks made of a different material that finally snapped after however many decades of use. Amazing that only three of them held it up for that long.

I swear, the replacement shelving is going to be so overdesigned....