Well since it was raining this evening and fixing the back was on my mind I went ahead and did it.
The plastic neck protector cup was easy to remove which both makes work easier and will likely lead to a better end result as it's mounting tabs on the inside would have prevented the fiber board from laying flat if I left it on. I started by laying it flat on concrete, and letting it get saturated in water which got it started straightening out. After letting it sit a bit; I then used the stub of a brick in the photo below as a weight to slowly work the the milder warps out and diminish the worse ones until it got flat and pliable enough to do what is shown in the photo below...Put a flat board(in a garbage bag to keep it dry) on top of most of the warped area, weigh that down with a tire(with water in the concave parts), and use some water filled milk jugs and that chunk of brick to weigh down the remaining parts.
The result is that it seems to be flattening decently.
I'm going to let it sit weighted like this outside overnight. Hopefully tomorrow there will be sun and it will dry and hopefully it will dry straight.
Once I have it working better, if I decide to restore the cabinet, I'll either lacquer the whole back or lacquer the labels(to preserve them) and paint the rest.
The reaction of the folks upon seeing the warped back before the straightening, was a shocked "What's that!?" from dad, and mom joking that one could put it on a stand and call it modern art, then observing that it is a bit moldy. LOL.