No personal experience with promtail, but ... have you tried redirecting the STDOUT / STDERR to somewhere?
/usr/local/bin/promtail -config.file /etc/promtail/promtail-local-config.yaml > /dev/null 2> &1
Above example will (attempt to) redirect STDOUT to /dev/null (i.e. "oblivion") and STDERR to "follow the footsteps STDOUT is going".
Of course, the script / app can try to forcibly write stuff on terminal, but doing that is generally avoided to maximize compatibility; it is completely possible that
there actually might be no terminal - it can be a true figment of script's imagination
the user might actually know better where to put any kind of output (personally I might just target STDERR to something like /dev/tcp/errorlogginghost/666
or ... whatever etc.