I'm essentially going to be the grammar Nazi within the grammar thread here, but there's a difference between grammatically (or syntactically) correct and semantically correct. The classic phrase is 'colourless green ideas sleep furiously' - a sentence which is entirely grammatically correct, but is still complete nonsense. It's syntactically correct, but semantically incorrect. To make a correct sentence, it has to be both grammatically and semantically correct.
As long as you conjugate the verb after it properly, then using 'they' is always grammatically correct, because that's what being grammatically correct means. The question is if it is semantically correct - are you using the words to mean something sensible and correct? That's the question at hand, not grammar, and it's that which is deciding whether using they as a singular pronoun is acceptable. And I think, given that the 'unacceptable' poll option only has 7%, then we're generally in agreement that it is.
The question of whether it's correct can often be answered by whether it 'sounds right' and I think this often depends on the verb that comes after it. To take an example from @itwastuesday's post above:
...they would only accept "his or her"...
This sounds fine, since the verb, 'would' is conjugated the same for he/she and they (i.e. it's both 'he would' and 'they would'). This makes he/she and they pretty much interchangeable. It's different for verbs where the conjugation is different (e.g. 'he has' but 'they have') but this is made easier with contractions - 'he's' and 'they've' become pretty much interchangeable too.
The problem really lies in writing, as you use fewer contractions and, since it's all written out, it's easier to pay attention to the verb. For example, consider the sentences:
He is a big guy. They are a big guy.
The second one, to me, sounds wrong. You're using 'they' to refer to a singular noun, and it just ends up not quite right, so the language isn't entirely there. But I feel like this is one of the cases that will become more accepted over time - certainly 'they're a big guy' is a lot less weird-sounding. And considering we're halfway there already, it's certainly a lot better than inventing a whole new pronoun to somehow get the entire population using and used to.
tl;dr: It's semantically incorrect to say that using 'they' as a singular pronoun is grammatically correct, but using 'they' as a singular pronoun seems to be generally semantically correct.