One way or the other, your hosts are running your application, and you are already deploying software on every single host. But I hear you with some of the agents. That's why I mentioned syslog.

It's already there, it's supported by most logging packages, and it's dead simple. No additional software required. All text. What it doesn't do is structured logging, but analyzing on the log host is often enough.

Agents aren't all that bad however, and you're likely already running some agent like icinga or zabbix for regular monitoring.