i'll agree, it should but put down, but you came off rather harshly.. Not trying to start a fight with you, and i agree with you 100% (personally i think any animal that is like that should...just my opinion) but yes, you did come off rather harsh...
X2 though. Put it down to stop its suffering, even if it doesn't seem like it is.
Definitely needs to be culled ASAP (if it hasn't been already) and I would also not breedfrom the mother again. Over grooming is a habbit, and a rat that's done it once, will likely do it again, and you will lose more babies.
If it were a pet rat that had care detailed to its specific needs, it could potentially live missing one leg.. but three of them. :/ Yeek, poor little guy. I misread this at first as "rat with one leg" not "one-legged rat" (boo dyslexia) but the picture cleared that up. :[