#!/bin/ash
set -e

# when no arguments are passed set options from environment
# otherwise append all arguments to the remote_syslog command
if [[ -z ${1} ]]; then
  exec /remote_syslog/remote_syslog -D --dest-host=${SYSLOG_HOST} --dest-port=${SYSLOG_PORT:-514} \
  	--tcp=${SYSLOG_TCP:-false} --tls=${SYSLOG_TLS:-false} --facility=${SYSLOG_FACILITY:-user} \
  	--severity=${SYSLOG_SEVERITY:-notice} --hostname=${SYSLOG_HOSTNAME} ${SYSLOG_FILES}
else
  exec /remote_syslog/remote_syslog $@
fi

#!/bin/ash
set -e

# when no arguments are passed set options from environment
# otherwise append all arguments to the remote_syslog command
if [[ -z ${1} ]]; then
  exec /remote_syslog/remote_syslog -D --dest-host=${SYSLOG_HOST} --dest-port=${SYSLOG_PORT:-514} \
  	--tcp=${SYSLOG_TCP:-false} --tls=${SYSLOG_TLS:-false} --facility=${SYSLOG_FACILITY:-user} \
  	--severity=${SYSLOG_SEVERITY:-notice} --hostname=${SYSLOG_HOSTNAME} ${SYSLOG_FILES}
else
  exec /remote_syslog/remote_syslog -D $@
fi

Add -D for command line case as well
Add -D for command line case as well

Without -D, the container exits immediately, and can't send logs effectively.