Forcing their customers to 1) move the mouse cursor over the search field box, and 2) click it before allowing you to type what you want, strikes me as an oversight. Enabling text input right away (as google does) seems like an obvious UI choice here. Especially when looking up a specific item is very likely the number one motivation to visit Amazon in the first place.
Now I am sure Amazon must have a ton of metrics on user behaviour. Maybe most people browse by clicking through categories. A lot of users probably just land directly on product pages via google results or affiliate links. But even if that is the case, having the search field ready for text input as soon as the front page loads will a) not hinder anyone from browsing while b) empowering everyone to spend their money more quickly. Instant win-win.
Or so it seems. What if that specific friction in user interaction is no oversight but attention to detail instead? What if two seconds before a product search is just enough time to peak your interest in an additional product advertised by a front page slider? What if their user metrics underline that to be true? Game. Set. Match. Amazon.