Mostly working solution: see http://jsbin.com/etovur/5 - when the date input gets focus set the value to '' after a delay.
setTimeout(function() {
document.getElementById('adate').value = '';
}, 0);
The biggest problem is that this workaround makes it hard for the user to pick today (the user must scroll away from today and back again to pick today). Since this is an effect of the IMHO flawed over-simplified UI of the datepicker on iOS, I don't think there can be any viable solution to my original question.
Some possibility of timing bugs (next then quickly next on slow iPad?).