Yes, the authoritative server get less data from the DNS server but the owner of the domain already get the information from web server logs and similar sources

Yes, the authoritative server get less data from the DNS server but the owner of the domain already get the information from web server logs and similar sources. I do not see how dns mixing provide any privacy in any common threat model. In the first case a client that request a webpage contact company X authoritative server with private information [IP ADDRESS], creating a record on the DNS server, and then visit company X web server creating a second record at the same company with the same [IP ADDRESS]. In the second case a client request a webpage of company X by contacting google, creating a record on google DNS server with [IP ADDRESS], and then visits company X and create a record there with [IP ADDRESS]. In one case one company has the data, and in the second case two companies has the data. It does not make any sense.