So, the common idea is that the UV filter helps to protect the lens element from damage. Stores love to push that concept, a lot, and I think it stems from trying to actually get rid of them in the digital age when it didn't really need them. They've now managed to create a sustained market for UV filters despite the fact the UV part of it is meaningless. Kudos to them, I suppose, but I think it's a waste of money and I'm not the only one.

So, in the scenario you just described, you may now be out the cost of the shattered filter and possibly the cost of a front element replacement, though this is not as expensive as you might think. Filters are not strong glass, unlike your lens element, and so it is entirely possible that had the filter not been there there would be no harm done (though you may have calibration problems, that depends). You may still not have significant damage, as a result of the filter, it takes a lot to make a lens useless, but you never know.

For me, I don't use them. A lens hood is far more effective in blocking significant contact with the front element. If there is some likelihood that I might be in a massive sandstorm or something, then maybe I'd slap on a UV filter, but probably not.

As further food for thought... A filter is another piece of glass. Cheap ones can degrade image quality, but all of them run the risk of light reflection creating ghosted reflections in your shots. When I start to add up the plusses and minuses, I'm not seeing a lot of plusses for the filter.