I agree with everything in your post except this part. What I remember from my biology classes is that a hand on a pan wouldn't need brain processing either. Perhaps a more appropriate example would be a gunshot going off to start a race. In that case, before you start running, you will have to process that the gunshot means go, and then send the message to your legs.
The hand on the pan does need brain input. Its simply that your brain and hands have such constant communication that the pathways are heavily trained and much quicker than you think. Burning on the sensory nerves is totally a cognitive function. If you're cooking on a hot stove and someone comes up behind you and puts an ice cube on your neck, your nerves send a cold impulse to your brain, but your brain often misinterprets it as heat and responds as if you're being burned.
It is also evident because when you touch a pan, it takes a brief time to process the heat. That is why we tap on things we think are hot. We test the waters by touching it quickly. The heat transfers at the same rate, but a quick touch requires the brain to process it. If you simply grab the pan to see if its hot, you'll fry your hand before your brain processes it and decides to pull away.
The gunshot also requires brain function, but a well-trained and experienced racer has trained his or her brain to fast-track those impulses.
The five senses all require brain function. Reflexes don't. If you have a recently deceased cadaver and burn it with a hot pan, it doesn't react. If you smack its knee with a hammer, the knee still jerks.