Central logging offers a lot of potential. Lots of reasons for it but just some...
Really easy to produce central reports to look for issues across the entire estate, not just server by server.
Easy to give log access to people who are not system admins.
Much more secure, you can look at logs without access to the systems themselves.
Works on systems that don't even have the ability to log into (like Windows Nano and DevOps systems.)
Encourages good tooling, like nice graphical interfaces, reports, etc.
Allows for correlation between systems, apps, etc. Compare a single app on many servers at once, etc.
Protects logs in case of system failure
Stores logs in a permanent way for ephemeral devices.
Increases the reliability of alerting.
Increases options for alerting.
Allows files to be backed up under a central policy.