There would be usually multiple syslog sink servers so there is never a single point of failure. This can be configured in various ways either dns round robin,

There would be usually multiple syslog sink servers so there is never a single point of failure. This can be configured in various ways either dns round robin, load balancers or even multiple destinations configured on the client side, rsyslog can iterate over a list in case of failures. 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