Deer don't regrow antlers the way lower animals regrow limbs

> They found a group of cells – called PMCs – believed to be crucial for limb regeneration in frogs and salamanders – to be abundant in deer antlers as well as in the tips of mouse digits.

So there's some positive evidence behind the mechanism being similar -- the same sort of cells are responsible for regeneration in deer and frogs. Which leads us to the question, how similar is this process? That was investigated by the scientists, who found that the source of PMCs to be different:

> However, the researchers also discovered that the PMCs in antler-generating tissues were formed by a sharply different process to lower-order vertebrates, according to the study.

The article continues in some detail as to the specific differences.

Antlers are bone, not keratin. During the growth process they have skin, blood vessels, and nerves.

"Each antler grows from an attachment point on the skull called a pedicle. While an antler is growing, it is covered with highly vascular skin called velvet, which supplies oxygen and nutrients to the growing bone. Antlers are considered one of the most exaggerated cases of male secondary sexual traits in the animal kingdom, and grow faster than any other mammal bone. Growth occurs at the tip, and is initially cartilage, which is later replaced by bone tissue. Once the antler has achieved its full size, the velvet is lost and the antler's bone dies. This dead bone structure is the mature antler. In most cases, the bone at the base is destroyed by osteoclasts and the antlers fall off at some point. As a result of their fast growth rate, antlers are considered a handicap since there is an immense nutritional demand on deer to re-grow antlers annually, and thus can be honest signals of metabolic efficiency and food gathering capability."

https://en.wikipedia.org/wiki/Antler

To summarize:

- antlers grow from the tip, while hair/fingernails/horns grow from the base

- antlers are living bone tissue, while hair/fingernails/horns are dead keratinous tissue

- antlers are made from differentiated tissues (cartilage, bone, skin, blood vessels), while hair/fingernails/horns are just a keratin matrix

But FYI keratinocytes are quite mysterious too. How do hair and nails grow only in one direction? That’s quite strange and nobody knows.

So if you think of the evolutionary pressure, there MUST have been some deer at some point that didn't shed their antlers. The question is: would a female choose a male that tries to be economical about it's resources and save the antlers for next year? Or would the female choose the male that can take that resource hit every year? It's kindof like asking present day women: do you prefer a male with a wasteful Lamborghini or a male with an economical VW or Toyota hybrid? I think we know what the answer is...

Also, biology has another angle to it, but I think this could have been overcomed evolutionary: the antlers grow form the top, so in order to keep the current antlers and then still grow them, the deer would have had to keep the velvet healthy. Being such large sharp tools, used for seasonal combat, keeping the velvet could be hard. I think this wouldn't have been un-surmountable if the deer chose this strategy, but I think my first explanation might be the most likely one.