That works fine until your centralized server falls over, either because you're scaling to hundreds or thousands of machines and your logs are chatty or someone goofs and starts logging kilobytes of data on every request.
If there is log storm coming from misconfigured app depending on your traffic levels the sink servers should give you a lot of space for an on-call engineer to be notified that something is off and adjust / rate limit / drop accordingly.
Edit: udated with more info