Looking at open-telemetry/opentelemetry-specification#3740 (comment), I noticed that the "console" loggers are using java.util.logging, which I suspect could get fed back into the logs "console" exporter if someone is using a jul bridge back into their logging framework and using our logging framework appenders.
I think #5918 could be used to address this (currently suppression is only checked for spans, so we would need to add a suppression check for logging instrumentation also).