Great article and pretty sharp analysis. The only thing that you did not take in account is the massive SEO power each of these links you removed have on the home page.
Everyone of the links that are present on the home page are there for a very specific reason and placed at a very specific place.
For sure, the UI is relatively old school and could probably be improved in many different ways. But the human brain has the faculty to scan and avoid the parts that we are not interested in.
As a matter of fact, once you already booked once, the next bookings are extremely obvious and pretty simple to achieve.
I wonder if booking.com audience is more recurring than new, but it would not make sense change any single pixel if more than x % of your audience is used to perform an action very well.
Anyway, there is always room for improvement… Always :)
Never stop optimizing!