Short chainstay length is one of those advantages indeed. As tyres got wider, traditional frames had to run longer chainstays to alow the chainstay itself to fit between the tyre and chainrings. An E-stay doesn't have that problem.
Chainsuck is still very much a problem. Without the chainstay being in the way, the chain can get pulled upwards until the lower end hits the higher end, or until the mech tensions up so much that it pulls it off the chainring again (or worst-case scenario : until the mech gets pulled to pieces)
Mud clearance is often claimed to be an advantage, but I do have my doubts about that. Sorry that it isn't all shiny, this bike currently does 300 miles/week.
Yes, there's plenty of room on top, but still none at the sides. These are 1.95" slicks, with the original 2.1 knobby affairs it's worse.
As for why they disappeared, I reckon there are a few reasons.
1 ) e-stays were a lot more nervous and require more skill than the more stable regular designs. That 1" or so difference in chainstay length makes a world of difference.
2 ) BB flex, albeit present in a lot of designs, is made out to be much worse than it actually is. It's usually caused by rider error (stomping on the pedals in too high a gear) too. Still, once such rumours start to spread, they are quickly mistaken for the truth.
3 ) Most manufacturers didn't get it right BITD. Sbike e-stays broke right behind the seat tube, Alpinestars ruined the Cro-mega and Ti-mega's strength with the 1992 redesign (they all crack in the bit of tube above the dropouts, just below the actual rear triangle), Al-megas and Aliens cracked just about everywhere, etc etc. The entire design got a bad reputation because of this.
Oh, and the single e-stay designs (like my Sbike) need special attention when driving with normal pedals. If you're not careful how you place your feet, those cantilevers can really hurt the inside of your ankles.