Just to be argumentative, the 808's circuit was designed the way it was because it fit the technology of the time and kept it cheap. It's nothing more than a damped resonant filter, something that in another context might be considered just a really bad oscillator. A very similar circuit is also used for the faux-toms. The clever part was how they got the attack, using a technique not far removed from the trick used in Hammonds to generate even-harmonic percussion.
 
The cymbal effect was indeed clever, though. And the handclap sound still stands up today. Not because it sounds like hands clapping, of course, but because it's a satisfying percussive sound. It can, however, be blended with actual clap samples (or snare) to nice effect.
 
Here's the actual 808 bass drum schematic, for the hopeless nerds in the audience...the meat of the circuit is op-amp U12.
 

 
I found the above image here, which also includes a pretty good overview of how it works, as well as how to emulate it in software.

Cool facts there, Mr. Flipper. Almost makes me want to buy a vintage 808 and warm up a soldering iron (almost).
 
I looked around for an impulse for the SSL talking gate/room. Couldn't find that, but I did find this article on recreating 80s reverbs. The picture inset at the end of step 8 sorta reinforces the point.

Ah, yes, my mistake. The 808 had only one resonant filter (and with pitch decay).
It was the TR-606 that had a second one for the beater part of the sound.