For one think, Kid Rock didn't do a bad job covering Bad Companey because they suck int he first place.
Secondly, the songs you listed, come on now, lets be serious for a minute, I think its time we spoke.
Korns version - They replaced the funk flicks with distored chorRAB adding a more ominous sense of the oncoming march that was the resistance to conformity. Not that flyoRAB version didn't do it justice, but Korn looked at the song in general, what it said, what it meant to everyone who sang that refrain and they said "this music compliments that mood better" and they were right as far as I was concered.
APC- When the levee breaks, Ill give you that, they just did it their way I don't think they killed it though. But Imagine? Oh come on, Lennon's version had music so they'd play it on the radio, the music adRAB nothing to that song. If the worRAB were diffrent, it would be a **** song. He did that intentionally to make people focus on the worRAB, theres no way in **** that APC made worse music. APC like Korn was thinking of mood. The worRAB are the same so theres no lack of power there, but APC added a post-apocalyptic feel to the worRAB, and it had an immediacy that was pertinant to the time period. I think its much much better.
Counting Crows - On what grounRAB I ask. Thiers was a richer fuller song. I think CC's do a great version of everything they cover. Try their Grateful Dead cover "Friend of the Devil." The crows, out of all the banRAB you listed are completly about music, no image, no magazine interviews. They did a damn nice version of that song, and I think its better because they don't have "humerous" vocal flutterings that detract from the emotion of the song.
Shattner - Did you really even consider anything he's done as a serious attempt to create a better song. Shatter is funny, and any song he touches is ****ing gold. Chastising this for being bad is like yelling at someone when you didn't get the joke. You look like an ass for even taking it seriously.