I'd just like to say that the Emirates' Identity Card website thingy appears to be working and I've just used it to book two appointments for myself and Sarah. They're both at Rashidiya in April, because Sharjah is apparently booked all year, but I'm not complaining. They're both on different days, too, because there were no contiguous appointments, but I'm not complaining. And the system just wouldn't buy a family application for two people, but I'm not complaining.
Now we can go an use the new and enhanced application application.
It looks as if it might go right this time. The site's here. I even got a text from them confirming the appointment a few minutes after I'd completed the form.