Assuming one's software allows for that. Which a lot of software either doesn't do, doesn't do with sufficient configuration, does badly, or does but doesn't support your particular logging method.

As an example, nginx only recently gained the ability to log to syslog; Apache has a logging module but it's not exceptionally customizable if you wanted to log to, say, ActiveMQ, or to a custom service (unless you write a separate binary to accept logs on stdin).

Yup. Given that those logging mechanisms also expose all the problems with file based logging, you have a much more significant effect with a lot less complexity by working on fixing those deficiencies than this route.