"Those translation queries are answered by our authoritative name servers that occupy well known IP addresses themselves, which in turn are advertised to the rest of the internet via another protocol called the border gateway protocol (BGP)."
"To ensure reliable operation, our DNS servers disable those BGP advertisements if they themselves can not speak to our data centers, since this is an indication of an unhealthy network connection."
Correct me if I am wrong, but here "DNS servers" means the computers, not the software running on them, i.e., each computer is running both DNS software and a BGP daemon. I am not aware of DNS server software that disables BGP advertisements but a BGP daemon could do it.
For example, a BGP daemon like ExaBGP can execute a DNS query, check the output and disable advertisements if the query fails.