Still seems awkward though.
I don't know. I used it for quite a while now and it seems completely natural for me ;)
I would argue with the "one" part in that text. "Every
one will have their tickets booked" is perfectly okay, so I don't see how "one" may imply its usage with "they" would be absurd. Besides, Shakespeare used "one" with "they", so why we shouldn't? :P
It's pretty complex however. At times I'd much rather use "his", whereas another time (especially in case when person at hand is a representative of a certain group) I will find "their" more convenient.