Regarding fingertips growing back naturally, I can cite a personal example - my mother. When she was a child, she came in after playing in the snow and slammed the door shut on a fingertip. This sliced off the flesh and some portion of the bone, but the nailbed must have been left intact (as with the linked article).
I'm assuming there must have been some metal strip along the door or frame, to slice it rather than just crushing it. Apparently her fingers were cold enough that she didn't feel it at the time, which I would assume also helped recovery. This was sometime in the 1950s, so definitely no magic pig's bladder powder then.
Her finger healed and subsequently appears completely normal. The fingertip is covered with flesh and skin, the fingernail is normal, and she has no loss of sensation. She's had a career as a piano teacher, so there is certainly no loss of dexterity either! But if she puts her hands together, that finger is noticeably shorter than the finger on her other hand by about 3/8", due to the loss of bone length.