Why are there so many 7700 squawks?
7700 is a "general emergency" squawk. It tells ATC that there is "a problem" of some sort with a particular plane. And I agree that this "general umbrella" type of squawk is a good idea. I have several disagreements with the FAA, but this is a good one. Several thoughts:


the specific nature of the emergency is only for the PIC and ATC. Nobody else needs to be informed, they can't help anyway.
oftentimes lost comm is part of the problem. So they can't talk, but you still want to show ATC that you have an emergency. You could squawk 7600 for lost comm, but if you have some truly serious, like an engine failure (I personally don't consider lost comm anything even remotely near "truly serious"), you can inform ATC with 7700 that you have a more serious problem than just lost comm. They'll notice the "lost comm" aspect of this situation on their own, by not being able to talk to you. Now they know you have something more serious, because you don't show 7600.
If you really want to use different squawk codes for different types of emergencies, it would merely add to the already large amount of numbers and codes pilots and controllers have to learn by heart. In an emergency where time might be of the essence, trying to remember abstract information like this can be difficult.


"The first time it happened, I thought the plane was about to crash"

Why did you think that? 7700 is not some "we're gonna die" type of squawk. It's a general "umbrella type" of squawk for all sorts of emergencies. An intelligent pilot will squawk 7700 for any "small" or "big" type of emergency (and who is supposed to be the judge on "big" vs. "small"?).

"How is an aircraft going down to be differentiated from one carrying a sick passenger?"

It isn't, because it doesn't matter. The people that are familiar with the situation (PIC and ATC) are in communication, unless lost comm, in which case the controller who can't glean more info from the 7700 squawk can't help anyway, and nobody else needs to be informed because they can't help anyway. Squawk codes aren't meant for desktop people like you who download some app and watch.

"Would you recommend handling this issue differently than it is done now, if you were in the position to?"

No. There are good reasons to have an "umbrella" type of squawk for all types of emergencies.