I'm currently reading logs via command line by ssh as you describe for a simple service and not satisfied with this. What would be a step up without a lot of extra infrastructure?
I’ve personally used Graylog[1] with success in the past.
However, I’ve had an excellent time with Grafana and have been following their Loki[2] project.
My company uses other solutions, so I haven’t needed it, but the Grafana stack might suit your use case.
Run vector [1] on each host to collect logs and send them to New Relic's free tier.
No need to manage extra infrastructure for logging, New Relic works pretty well and also has alerts, and Vector makes it easy to move to a new provider or self-hosted solution without reinstalling agents: just add a new sink to your config file.