I can confirm this; I'll have a look in Safari dev tools later, but really the only thing we changed here was to remove the results attribute but as you say other inputs seem to behave fine so it may just be some random quirk.
If any one can test on iOS 9 Dev Beta 5 (@Jeremy ?) then that'd be appreciated.
Interestingly it doesn't seem to work in iOS 8.4 either, but as Jeremy says above, it appears as though in the latest iOS 9 beta it works fine.
I can understand where it might get confused. The search box, in theory, should focus when the search icon is pressed (and it does on Chrome and Firefox). It would suggest that the automatic focus is possibly happening without bringing up the keyboard, so it then requires a second tap.
We've done some more testing on this today and there isn't a completely apparent fix. It's something we think is worth fixing, but the only fixes we've come up with so far are really off the wall - the kind of thing that really shouldn't need to be done.
So, I'm still hoping this will begin to stabilise a bit at least over the next few iOS betas. There is some more iOS 8.4.x releases due, too, so they may contain some sort of fix (the issue is apparent there, too).
Essentially, no -- or it will at least potentially need some bigger changes. There are limits on what "focusing" means on touch-devices when it's programmatically triggered (when it isn't technically in response to a click).
We have reproduced this, but it may not be something that can actually be changed due to these limits.
We believe we have identified a workaround to clicks on the input not focusing as expected. It's actually related to hiding the search icon that brings up the search menu. Very strange. Good catch @Chris D
I would note that due to general mobile limitations, we can't actually autofocus things like this, so a separate click is generally required. It does focus, but it doesn't bring up the keyboard. This sometimes puts them in weird states.