Plywood front would be harder to make and install than just stapling up some fabric....but easier in the long run.
Watch for loose/unraveling threads on fabric, they may try to eat them or get tangled up in them,....and staples coming loose.
It's not about 'privacy'...that's a human concept...it's about safety, they want to lay their eggs in a place 'safe' from predators.
A curtain or smaller entrance to nest can provide that illusion...little do they know that we will depredate their eggs for ourselves :D <evilcackle> hahaha
Changes can take time to be accepted by chickens.
Liked the old metal round hole nests, but way too big and expensive for my coop, so I emulated them in wood.
I have back access to gather eggs tho, might be harder in your situation.....
......just putting a 4-6" board across the bottom front...
I think the problem might be that the nests look no different than the floor of the coop.
I'd put a front on the nest bank with a couple or three 10" diameter holes (you could probably make 3 nests out of that length).
Or at least put a piece of wood along the bottom about 4-6" high.
Might try...