Interesting! this is something that I have thought over often. My own experience of log aggregation is limited to ELK stack and Loggly (for a brief time) where the setup worked fine but the workflow didn't. We just stopped browsing logs after a while. Although, a giant centralised system for logs sounds incredible convenient, making sense of them starts to become a huge problem and then, it's just easier to ignore log system.
I am sure the solutions discussed here have features to overcome this (filters / alerts) but IMHO, we'd be better of collecting less things — limited app events that have fixed formatting and are easier to make use of in debugging and monitoring.