It seems somewhat different to me from “the driver saw the person, and then ran them over anyway”.
It is of course not equal to whether the driver was at fault or not. The driver could be at fault or not in various circumstances where either they did or didn’t see the person hit. E.g. If they didn’t see the person because they were texting at the time, or looking at their spedometer to insure they weren’t going over 20 mph in a school zone, which caused them not to be watching the road.
However “the driver didn’t see” is I think often used as a shorthand summary of the gist of a situation. “The news” (aka entertainment industry) in particular tends to favor one-liner sound bites over accurate detailed statements.
And yes, “I didn’t see them” could also be an attempt to avoid responsibility. Or it could be a natural and fair statement about a more or less unavoidable accident by a generally safe driver.