Some good observations. Here are mine.
Works of art are often as much about when they were created as what they are, as that informs how they are made. Just because times have changed doesn’t mean it’s impossible to recreate things as they were from the past, but because you're not in that original moment it's easy to miss some of the ineffable qualities of the work being simulated. The result—as in the Tom & Jerry—example is simulacrum; having neither quite the character nor the innate sense of the original, and without having a clear sense of personality and style of its own and ergo ends up a crazy quilt of elements that don’t gel; a pastiche in the “confused mixture or jumble” sense.
The example cartoon fails to work not because it’s too faithfully recreating a bygone era and style, but in part because the filmmakers are only copying the obvious aspects of the originals but missing other key ingredients of the mix. At the same time they are failing to grok exactly what made the originals work in the first place. They copy types of gags from those 40s-50s MGM cartoons, but their gags aren’t as funny because they don’t have a firm and sure grasp of things that are harder to copy: such as a feel for comic timing (how many frames do you hold on something, how fast something moves, etc.), and how are these elements built up for maximum effect.
For instance, the basic premise is a Tex Avery staple of one gag that is riffed on for dozens of escalating variations, but the film detours off of that as the filmmakers try to cram it full of elements from other cartoons, so much so that it hurts the main thrust of the film because it doesn’t build properly.
To bring this back to the subject of fan films, I think the lesson here is that you can slavishly recreate the settings, sound and visual style of a show and yet fail to grasp that the show was more than just those few elements. So what typically emerges is something that looks and sounds like the original, but without the same soul. To use a sci-fi cliche: it's usually a pod-person/production.
While the John K. and Bob Camp and South Park examples are interesting, neither of them is trying to recreate something from the past. John K. is a huge fan of Bob Clampett (moreso than Avery) and he often applies some of Clampett's aesthetic to his cartoons, but he's not trying to copy Clampett. Hell, even when he works with established characters like Yogi Bear, he often parodies the original style rather than imitating it. That's a different—though perhaps related—animal.