Opening up google.com I see that the search input field gets focus automatically, i.e. the cursor is already blinking in the field and users can start typing right away.
In contrast, amazon.com or booking.com don't do this. Yes, 99.9x% of the Google users will search and appreciate that they can start typing right away, while at Amazon or Booking there is a chance that users want to browse the site instead of searching right way. Anyhow, would it hurt to give the input field focus and make it convenient for those who want to start typing?
Q: What are reasons NOT to give a search input field focus?
After reading a related older discussion here on the site about why Google would not do this, I learned that Accessibility might be a reason. So I'd be very interested if there is an accessibility-friendly way to work with autofocus?