Dongl – I’m not familiar with synology syslog, but I was just toying with Zeroshell’s syslog so thought I would add some info in case it helps you.
zeroshell’s version of syslog does not seem able to send messages over tcp (haven’t poked at enough yet to verify for sure), so that might be your problem. my system was initially only set to receive tcp.
also if possible run netcat on your synology server and see if any messages are coming through from zeroshell (or some other way to do packet sniffing on your network)
nc -lu -p 514
this showed that zeroshell was indeed broadcasting messages for me.