How do I get the remote hostname of the server?
I don't think it's possible. I think what you're looking for is gethostname(). That's not called anywhere in the PostgreSQL source code. Nor is utsname or any kernel interface into this.

What is called is getaddrinfo that provides a CANONNAME but that's essentially just your own method of addressing the server not the "symbolic hostname". If you need the hostname, you'll have to build a C-extension that does just that. Or, use a third party procedural language, like perl plperlu, that has that support (through Sys::Hostname or Net::Domain, see perldoc -q hostname); or, alternatively through javascript with plv8 that has os.hostname().

Another method would be simply reading the file /proc/sys/kernel/hostname.

It's a database, not a remote administration utility. You can't really fault it for not doing this.