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You say the not-so-good tree grew on its own? Volunteer fruit trees that grow from seed usually produce fruit of inferior quality because quality fruit often does not breed true. The tasty fruit tree has been cultivated for years, and possibly grafted onto sturdier root stock. If you want another fig that bears nice fruit, your best bet is to plant a new tree. This article has some ideas: https://www.gardeningknowhow.com/edible/fruits/figs/different-types-of-fig-trees.htm

I think the best thing is to prune the dead and weakened branches out of the top of the tree. Don't worry about fruit too much this year; you want the tree to put its energy into growing new branches for the future. Read these tips on pruning fig: https://www.gardeningknowhow.com/edible/fruits/figs/pruning-fig-trees.htm

Yes, it sounds like it was damaged by the winter. If you prune the branches back, the tree will ultimately be much healthier. Here's an article on fig pruning: https://www.gardeningknowhow.com/edible/fruits/figs/pruning-fig-trees.htm
Also, some tips for winter protection: https://www.gardeningknowhow.com/edible/fruits/figs/pruning-fig-trees.htm:

Your tree is not dead, because it's sending out new shoots, but it is traumatized. However, with TLC and some pruning, it should recover. The first thing is to cut off the dead wood. You can start with the smaller twigs, starting at the ends and working backward until you come to green tissue. You can scrape the larger branches with a knife to see if there is green under the bark. Here's an article on fig pruning: https://www.gardeningknowhow.com/edible/fruits/figs/pruning-fig-trees.htm
and here's one on fig care: https://www.gardeningknowhow.com/edible/fruits/figs/how-to-grow-figs.htm