In fact, I would reverse SaaS entirely: make it Service as a Software instead. That is to say, provide the service you want manually for the customer (as much as is possible, as it is not always possible), then build software to automate that process.

When DoorDash first started, they manually took orders and delivered them to people, over the phone. They'd get an order from their customer, then they'd order it from the restaurant, then drive over to pick up and deliver them. Then, they built the driver network software, mobile app, and all that.