singular "they": a practical objection
[We should be trying to communicate clearly, and sometimes language rules prevent that.]
I agree. This is why, when conventional language rules would dictate something that would make my writing harder to understand, I violate those rules. For example, I only place terminal punctuation inside a closing quotation mark if it is in fact part of the quoted text, because to do otherwise misleads the reader and is logically incorrect. That's not how the language rules evolved, but (fortunately) that's becoming a more common practice within the field of technical writing, and eventually we may be able to drag the rest of the English-writing world along with us.
This argument does not apply to singular "they", however. Or if it does, it doesn't apply the way you think it does, at least for some readers. If I see a well-crafted sentence that completely avoids the problem, I don't find myself thinking "wow, that was really unclear; he should have just said 'they'". Because it's well-crafted, I don't notice. That's good; one of the jobs of technical writing is to get out of the way so people can understand what you're writing about. On the other hand, every time I see a use of singular "they" that (I think) could have been easily avoided, it derails me in my reading -- exactly as an incorrect "it's" does. It distracts me from what I was doing -- absorbing communication -- and draws my attention to the writing itself. Further, that attention is negative; it lowers my opinion of the author or company whose work I'm reading. None of this is conscious and I can't will it away. I know I am not the only such reader.
While we should not necessarily write to the lowest common denominator, if one choice results in clear communication to everyone and another does not, we should follow the one that does, even if it's a little more work on our part. So quite aside from the (very real) religious arguments against singular "they", I hold that there is a practical reason to avoid it: it derails some readers and is not necessary.

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I suspect that it will eventually make the leap, just for lack of a suitable substitute. Yes, sentences can be re-worked to avoid the need for a gender-neutral singular pronoun, but it's a hassle, and people will keep looking for ways to fill the linguistic gap. So far, I have yet to hear a better contender than "they."
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People can tolerate all sorts of errors and glitches in spoken language that would grab their attention in written form. Speech is transient, and that makes a big difference.
English actually has a singular gender-neutral pronoun, but there is a cultural stigma against using it. One of my co-workers put it this way: "If the user doesn't like this, it can bite me". :-)
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As for "it": it's amusing that this option is rarely even mentioned. I think you're right about the cultural stigma, though I'd say it's more accurate to call it a connotation of the word: "it" is used strictly in reference to non-socially-recognized entities, be they inanimate objects, animals, or small children. The distinction is particularly clear in reference to animals and children, where "it" may be used to refer to strangers (psychologically, non-persons), but it is normal to shift to "he" or "she" once the speaker gets to know the subject. Because of this, any attempt to use "it" to refer to an adult person inevitably implies a devaluing of that person. That barrier makes it a hopeless case, in my view; any proposed gender-neutral pronoun needs to sound more polite than the existing options, or it will never be accepted.
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Depending on both my audience and the context, I'll use gender-neutral neologisms, established gender-neutral pronouns (one has more than just 'it' at one's disposal, after all), rephrase to avoid the pronoun, rephrase to a less formal second-person construction, go with the formal but (sometimes acceptably) clumsy 'he/she' or 'she or he', or simply grit my teeth and write 'they' anyhow.
If I know my reader s familiar with the same neologisms I'm most comfortable with, I'll write to hir using the same gender-neutral pronouns sie might use hirself. I may also use 'sie' and 'hir' if I think my audience will at least understand and I feel the situation warrants more concern for precision than for my audience's comfort ... or if I'm dealing with a special case involving a real or hypothetical subject of unconventional gender, but the farther I get from ssbb (news:soc.support.bondage-bdsm) and asb (news:alt.sex.bondage) the more weight I give to the more conventional options.
If I happen to know the person I'm writing to prefers another set of pronouns I'm at least familiar with, I may write to em using the words ey prefer, but stopping to think about it will slow me down more than rephrasing to avoid pronouns will. (Similarly for 'ze'/'zir', etc., of course.)
But all in all (and despite wishing sie/hir or ze/zir were commonly understood and accepted (though I don't particularly like ey/em)), the more I write paying attention to gendered/genderless pronouns, the easier rephrasing to avoid having to rely on the neologisms or the despised singular 'they' becomes. Of course, it does help to not be afraid to write "one" occasionally.