I just tried it and notice the same, so probably the backspace/delete key is being handled differently so focus is not needed, but then when you have focus on the field its conflicting with that behaviour.
However I don’t think this is an issue really? Clicking the X does not make it focus, so its only if you specifically click on the field that it will focus. Being able to edit a search (i.e. the only reason I can guess where you would need to focus the field) is not really important on a quick search feature whereas the functionality without focus is more important.