Even if the front element gets a few scratches on it, the effect on image quality is negligible. This is because light from each part of the scene strikes every part of the front element (unless something is causing vignetting). Although hard to believe, it really takes a LOT of damage for image quality to be noticeably degraded. Check out these pictures taken with a "slightly damaged" lens!
The assumption that any impact that shatters a thin, flat filter would have shattered the lens' front element had the filter not been installed is a false one. Most front elements are much thicker than flat filters, made of more resilient materials, and shaped in a way that gives them more strength (compare the shape of most front elements to the shape of an egg). Just because the filter shattered does not prove the front element would have. It can take a lot of abuse to scratch, much less shatter, the front element of most lenses.
One must also consider the possibility that a shattered filter and the resulting shards of glass and sand-sized grains can actually contribute to more damage than might otherwise be the case. This question concerns a filter that was shattered inside a camera bag that filled the internal parts of a very nice lens with the equivalent of sand. The lens had to be sent in and disassembled to clean out all of the tiny pieces of shattered filter that had gotten inside the lens.
Of greater concern when a lens takes a dive is the alignment of the elements in the lens. Some lenses have a reputation for tolerating abuse fairly well, such as the Canon EF 24-105mm f/4 L IS. Others, like the Canon EF 24-70mm f/2.8 L, have a reputation for getting out of alignment at the slightest bump. Often this is due to the way a lens is designed. In such a case the filter makes little difference. The ring will receive the lion's share of the force and transfer it to the front of the the lens housing that it is screwed onto.
A much better way of protecting your lens from impact is through the use a of a lens hood. Not only will a hood often keep objects away from your lens' front element, but in the case of heavy impact the plastic most hoods are made out of will flex and soften the impact force transmitted to the lens itself. And hoods contribute to better image quality in certain environments. A "protective" filter, on the other hand, can only reduce the optical quality of the lens.
On the other hand, when shooting in daylight, UV from the sunlight passing through your lens will inhibit the growth of fungal spores that may be (hint: are) in your lens¹. The filter stack directly in front of the sensor has a fairly strong UV filter anyway, so there's no real benefit to preventing UV light from entering your lens unless it is constantly exposed to sunlight 24/7/365, and there's a definite benefit to at least periodically allowing UV light through the lens.
For a fuller run-down of the various questions here that address the protective filter/no protective filter decision, please see the multiple links included in this answer to is uv filter a must?
¹ Fungal spores are everywhere on the surface of the earth, including the skin of most animals and the leaves of most plants. They require three things to multiply: Moisture, organic matter (provided by almost every kind of dust), and the absence of UV light.