People rarely realize that making a hotel booking at an OTA (mostly reduced to Expedia Brands and Priceline Brands these days) is a combination of their systems that then call the hotel systems to make the actual booking (or sometimes just a fax or email if the hotel has no IT). So information that Booking collects is mostly sent to the hotel system; there isn't any other way. Booking could add something to their side which is passed only to the customer (such as an ID of some kind) to ensure anything coming from them is legit since a leak of a hotel system would not have that. That would not help emails purporting to come from the hotel however and is likely doubtful to help much. I doubt most hotel brands would accept an email relay, they are already discounting the price to provide the OTA profit margin so not getting the actual email would be a sticking point.

Having worked at an OTA (before Expedia got us) I refuse to book at anything other than a legit hotel system. OTA's are fine for price discovery but you often get a better deal (and better service) from the hotel/brand directly. The front desk knows you booked via an OTA instead of the hotel directly.