Great study Filippo, I found it interesting reading.
In regards to the urgency messaging (13 people booked today, 30% discount today etc.) I have done a lot of research on this topic in my work. It’s true, these message types lead to large increases in bookings from customers. Booking.com certainly over-indexes on the types and styles of this urgency messaging (I think they provide anxiety more than urgency!). Some listings on Booking.com will be cross-listed (particularly hotels) across the web, so it is crucial to them that you continue your booking in their app – hence why they need this effective messaging.
This is a great example/ case study of where the real business needs collide with the designers vision for the user experience. The solution is to find a way to engage the customer, understand their intent and convince them to commit to the booking — all while ensuring the business sees bookings increase.
It’s just one of the micro-challenges in designing in this industry and another problem waiting for a designer to attempt to solve ;)