launcher logs is simply a wrapper for docker logs, which can show timestamps with -t. The following should give you timestamped logs:
sudo docker logs -t --tail 1000 app
The lines you quoted in those two files are not related to Discourse; if you figured so because there’s discourse_app in some of those lines, keep in mind that is a hostname and not a label indictating they are somehow related to the Discourse Docker setup.
Based on them, it looks like logrotate is failing, but you won’t see a detailed error unless you install a local mail setup. Assuming a Debian derivative, do apt install default-mta, then wait until another error, then check local mail with the mail command.
I’d also check server health like available disk space, memory pressure, etc. Extreme memory pressure (ie. paging) is a likely culprit for recurrent non responsiveness.