Agreed, if it's the only thing keeping you awake, but plenty of people find it easier to concentrate on a task if there is also a "distraction". It's like when school children fiddle with a pen to help them concentrate. The teacher might think it's a sign that they're not paying attention, but actually it's enabling them to pay attention.

I see it as a scale, where having the radio on or not could push a few point more regarding engagement in the task.

The same way someone shouldn't be driving if they're dead tired, but even if they're in an alert state, drinking a cup of coffee can sharpen them a bit more.