Otdfbt

I want to forward *.* to remote host via TCP/IP.

Also, I have local0 facility where messages are plain JSON messages and they has to be forwarded to same host, but other port (and uses same certificate for gTLS).

I've made a config:

# provides UDP syslog reception 
$ModLoad imudp 
$UDPServerAddress 127.0.0.1 
$UDPServerRun 514

# provides TCP syslog reception 
$ModLoad imtcp 
$InputTCPServerRun 514

$template logFormat,"[1234] <%pri%>%protocol-version% %timestamp:::date-rfc3339% %HOSTNAME% %app-name% %procid% %msgid% [type=syslog] %msg%n" 
$template logJSON,"{ "token": "1234", "env": "testfield" , %msg:2:$:%n" 
$WorkDirectory /var/spool/rsyslog # where to place spool files 
$ActionQueueFileName fwdRule1 # unique name prefix for spool files 
$ActionQueueMaxDiskSpace 1g # 1gb space limit (use as much as possible) 
$ActionQueueSaveOnShutdown on # save messages to disk on shutdown 
$ActionQueueType LinkedList # run asynchronously 
$ActionResumeRetryCount -1 # infinite retries if host is down 
$DefaultNetstreamDriverCAFile /etc/ssl/certs/AddTrustExternalCARoot.crt

*.* action(type="omfwd" protocol="tcp" target="listener.example.com" port="5001" template="logFormat" StreamDriver="gtls" StreamDriverMode="1" StreamDriverAuthMode="x509/name" StreamDriverPermittedPeers="*.example.com") 
local0.info action(type="omfwd" protocol="tcp" target="listener.example.com" port="5005" template="logJSON" StreamDriver="gtls" StreamDriverMode="1" StreamDriverAuthMode="x509/name" StreamDriverPermittedPeers="*.example.com") 
local0.* /var/log/app.log

Unfortunately, rsyslog doesn't even try to make a connection.
There's no evidence in netstat -nt, nor tcpdump

I'm looking for a way of forwarding those logstreams to their destinations without hacking it deeper in ELK stack. Can You help?

I would like to write this as comment, but do not have sufficient reputation.

Make sure the permitted peer matches the CN in the reciepients's certificate and that the CA used for signing is included in AddTrustExternalCARoot.crt.
You also do not have a $DefaultNetstreamDriverCertFile and $DefaultNetstreamDriverKeyFile set, so rsyslogd has no certificate to provide for client authorization.

I am assuming you edited the StreamDriverPermittedPeers="*.example.com" before posting and have a correct domain configured instead of "example.com".

It might also be necessary to have a matching entry in /etc/hosts.