Does the Mario Starter Course require a smart device?
Yes, a smart device is required. The box says the following:

Smart device required but not included

However, this set is not like Boost, where the intelligent brick just adds weight to your model without a smart device. Mario functions without being paired to a smart device. He can complete the course, collect coins, etc.
The most significant limitation without a smart device is that you can't update the firmware on Mario. The current factory firmware is fairly limited. It was missing several of the sound effects present in the current firmware, but most importantly, the factory firmware was able to recognize elements in the starter course only. Any elements from expansion sets were not recognized, and Mario would display the Bluetooth icon indicating that a firmware update is needed instead of taking the expected action:

This is resolved by pairing Mario once with a smart device. These features are then permanently available without the need for a smart device to be present.
Another important limitation for many sets like this that require a smart device is that there are no paper instructions in the box. Instructions are certainly helpful, but these sets are meant to be adjusted and rebuilt. The courses are made of very simple elements that you can likely put together by looking at the box. This might be challenging for a small child, but if an adult is willing to help out, it is possible to put these together without the on-device instructions.