Installed the latest 3.2.1 agent on a fresh and updated Ubuntu 16.04 amd64 .

The zabbix-agent doesn't start without throwing any errors.
I'm not an expert but I found out some issues with
/lib/lsb/init-functions.d/40-systemd

So it turns out a LSB log functions problem, I had to comment out some lines disabling log daemon in /etc/init.d/zabbix-agent (see attachment).
The /var/log/zabbix/zabbix_agentd.log is filled.

Maybe /lib/lsb/init-functions interaction should be checked.