Stables worse than lab... ooff, or should I hooff?? that's a rather large error.
I don't think so. Stables almost always force you into some kind of Engine/BM-Hybrid, which is most likely worse than a good engine. In the endgame, you're likely to have 1-2 Stables, but no treasure cards in you hand. Then Stables is just a dead card. Or you have to discard gold into a Stables, but what are you hoping to draw? Stables+Copper+Province? Surely not.
In the beginning, Lab generates more coins, but cycles more slowly. So Stables might be the better option. Later on, I can keep buying Labs as any Lab will probably make my deck better (aside from opportunity costs), but Stables have the potential at making your deck actually worse.
On the other hand, Stables are pretty good when you're being junked by Coppers (Mountebank or Jester could do this).
Both cards are good ones, but I like Labs a bit more.
From my experience, Soothsayer excels in games with no trashing or weak to no engine potential. Player playing BM or slog really only needs to buy 1-2 Soothsayers and VP cards, maybe with occasional Silver. His main economy - Golds, will come from playing SSers, so he doesn't even need to waste a turn for buying them. Of course, they become terrible in the endgame, where neither your Gold nor opponents' Curse will see the reshuffle and opponent's extra card may make the difference between $7 and $8; but by that time, your deck is already filled with Golds anyway.
I'd be very curios to see simulators' opinion about Witch+BM vs Soothsayer+BM to see if the latter wins.
Well, how often will you play those Soothsayers? Probably no more than 3 times, maybe less. I'm not sure whether 6 Gold would make a pesky slog deck so much better, especially if there's another good junker (any witch, but also all those "Ruins dealers").
And I'm pretty sure doubleWitch beats double Soothsayer like 70-30. The Witches will be played more often and they increase hand size.
edit: Would be nice if the simulator had implemented Soothsayer as a card. Doesn't seem to, though.