I've never had the automated tool do anything. It's too dumb to handle these types if itineraries.
Sometimes I've had a concierge proactively reach out, and sometimes I contact them. But I don't contact them unless there's a specific course of action I need them to take (i.e. rebooking me, letting a gate know that I'm running for it, etc.). In my experience, there isn't much they can do to help you make a connection. I tend to run faster than they do