When do you bring in message queues for more than just simple task offloading?

We all usually use message queues for basic stuff, like background tasks or sending notifications. But when do you really think about using them for your main system interactions? Im talking about things like event-driven setups between microservices, or managing backpressure better than just dropping requests. Are there particular signs or system traits that just scream "get a real message queue in here"? Or is it more about planning for growth and separating things from the get-go, even if simpler ways work fine now?