Wrong and poor advice to subvert browser functionality. Alert the user to which input they are currently working with. Eliminates [tiresome] user guesswork and is conventional. Also helps the visually challenged.
> via overlay boxes
I don't even.
> I enter my city and hit ‘Enter’. That triggers the search, without giving me a chance to enter my dates first.
It occurs to me that this is part of the design. They show you the inventory before asking you for your size, even though if you had the patience you can input your desired dates on the homepage. Also, note the date input is also available after form submission. Something tells me that they've done some research on user behavior.
> How else can Airbnb improve?
I'm going to stop now because if anything, this post is novice-level and poorly thought out if one is assuming that they can make better what they don't understand.