Most of the people I talk to accept 'they' as natural English. My highschool English teachers would probably be an exception, as was I until I decided to let it go. Wnoise probably has a point that 'singular they' is a matter of dialect, with most, perhaps unfortunately, having lost some of the more elegant subtleties.

And we can foresee that it will increase the amount of ambiguity in English even after this period of confusion is over. Is 'he or she' really so bad that this costly reform is worthwhile?

A good question. I'm happy to leave it with 'singular they' for most people but 'he or she' for people who want to signal sophistication (by speaking correctly). It is probably too late to hope to gain much relief from ambiguity except when you are familiar with your audience's manner of speech.

EDIT: I missed the great, great grandparent about singular bound vs singular referring. Thanks Wei.