Where does the privacy advantage come from? Once you resolve a hostname privately, don't you still need to use its IP address publicly for your traffic to be routed there?
That depends on your level of access/monitoring: if it's just a firewall log that shows source/destination IP, then yes - but as soon as you have any kind of packet monitoring, then the domain can be easily sniffed from the SNI header.
I think with https the results were something like 90% of page access could be guessed using meta-data (see eg https://web.archive.org/web/20090308103611/http://sysd.org/s...). That's going to drop when you don't know the site but you're going to need more counter-measures to hide your access effectively.