I purchased this lovely 3-caudex ponytail palm from a local reputable nursery. I notified 2 of the palms pushing new leaves up from the center, but not the third palm. Upon closer inspection, I noticed someone had cut the leaves of the new growth. The nursery was unable to tell me why. My question is, will the plant still push up the cut leaves and continue to grow, or is it toast?

It will. There will be a long period where you will see the cut leaves, but eventually it will grow out.