I think this is a simple matter of a wrong analogy on OP's side. They don't understand that "urgent" in this context means "that needs higher priority than ever

I think this is a simple matter of a wrong analogy on OP's side. They don't understand that "urgent" in this context means "that needs higher priority than everything else". So the saying makes sense: when everything is urgent and has equal "urgent" priority nothing stands above anymore. Being burnt out doesn't need a frame of reference other than the internal one of the "burnee". So it can be valid under any conditions.