I think you mis-characterize two things. First you mis-characterize skeptical theism. Second you presented epistemic skepticism too weak.  Skeptical theism is the view that denies the following inference:

Where S is an epistemic agent, if S can’t think of any God-justifying reason for permitting some apparent horrific evil, then it is likely that there is no such reason.

Usually skeptical theist point to the large gap between "God knowledge" and "Human Knowledge", capitalizing on the fact that we are fallible knowers, if we are knowers at all. Most skeptical theists are local skeptics, meaning they are only skeptics about particular premises or propositions. This means they are not global skeptics, they do not argue that most people don't know anything (the brain in the vat arguments are arguments for global skepticism, so it's kind of odd that you would present these arguments in conjunction with skeptical theism). Presumably your critique is that theists pick and choose which premises to be skeptical about, but they certainly are not using any argument like "proposition p contradicts theism, therefore i should be skeptical about p", nor does your critique mean that skeptical theism is false. At least most skeptical theists i've read don't present any argument like you mentioned; they just don't find the above inference very plausible, perhaps for the same reason atheists don't find the following premise very plausible:

Where S is an epistemic agent, if S can't think of any atheist explanation for the apparent fine tuning of the universe, then it is likely that there is no such explanation.

Last, epistemic skepticism I think is much stronger than you had presented. Epistemic skeptics often use the following argument:

(1) If S knows that S has hands, S knows that S is not a brain in a vat.

(2) S doesn't know that S is not a brain in a vat.

(3) Thus S doesn't know that S has hands.

(4) If S doesn't know that S has hands, then S doesn't know a lot.

(5) Thus, S doesn't know a lot. (3, 4) 

(2) in the first argument usually has a defence along the lines that there is nothing in S's experience that could suggest that S is not a brain in a vat. In short they agree with you that S can't even in principle know that S is not a brain in a vat.

In defence of the very first proposition, it's quite simple: If S has hands, then S is not a brain in a vat.

Surely you would not deny this. But if S knows p and S understands that q follows from p then S knows that q (this is a closure principle, that is widely accepted today).

Now we argue as follows:

(6) Assume S knows that S has hands.

(7) But if S has hands, it logically follows that S is not a brain in a vat.

(Eight) Thus, S knows that S is not a brain in a vat. (from closure, 6 and 7)

(9) Thus, If S knows that S has hands, then S knows that S is not a brain in a vat. (6 - 8) 

So either you deny closure, and thus (1) or you deny (2). Presumably you accept (2), since you argue that there is no way you can even in principle verify that you are not a BIV, that's fine. So some how you deny (1). Now i'm interested, do you deny closure? Doesn't it seem odd to argue that if I know something, say F, and I understand the logical entailment from F, I don't know what logically follows from F?

Also the great lottery problems are usually used in defence of Epistemic skepticism, for a good overview of the arguments see John Hawthorne's book on the subject. 

Finally with respect to why some people are epistemic skeptics, a lot of epistemologists suggest that knowledge requires what is known to be true. So, if S knows that p, p must be true.

Now listen to this statement: "I know that p, but i might be wrong". Doesn't this sound just awful? Some epistemologist think that it does sound awful so they suggest that knowledge cannot fallible (ie we can't have knowledge of p, where p might be wrong, or that the knowledge of p might be wrong), they say that in order to have knowledge we must have certainty. But if knowledge is not fallible, then we don't know a whole lot of anything. Thus we don't know a whole lot of anything.