You'll need to disable imklog or rsyslogd won't start:
module(load="imklog") # provides kernel logging support
This is essential unless you give full privileges to your container.
Here is a full working rsyslog config in Ubuntu/Debian:
module(load="imuxsock") # provides support for local system logging
$ActionFileDefaultTemplate RSYSLOG_TraditionalFileFormat
$FileOwner root........