It's UDP only. They also hard-code the port instead of looking it up with getservbyname, which is a little weird. If you want to use TCP, send syslog messages to localhost, and have a local rsyslog/RELP forwarder send messages remotely.
getservbyname under glibc uses dlopen, effectively working around static compilation. while this is a non-issue on ubuntu, the nginx devs are aware of the number of people building against uclibc or musl to produce static binaries for embedded use. I've seen nginx running on bare metal -- just it and libc, no kernel underneath. kudos to them for making this easier than they have to.
