While the container configuration includes the network_mode: host option that appears to simply allow the container to access to the container host's network directly and does not effect how host names are resolved. Since systemd.resolvd is not running in side this container DNS name resolution methods appear to be handled by /etc/nsswitch.conf and which currently suggests that only files and dns ("hosts: files dns") be used to resolve host names and IPs. In order for the browser block to resolve via mDNS it would either need to run systemd.resolvd, which understands mDNS or include mdns4_minimal/mdns in the /etc/nsswitch.conf file.
While the container configuration includes the network_mode: host option that appears to simply allow the container to access to the container host's network directly and does not effect how host names are resolved. Since systemd.resolvd is not running in side this container DNS name resolution methods appear to be handled by /etc/nsswitch.conf and which currently suggests that only files and dns ("hosts: files dns") be used to resolve host names and IPs. In order for the browser block to resolve via mDNS it would either need to run systemd.resolvd, which understands mDNS or include mdns4_minimal/mdns in the /etc/nsswitch.conf file.