Reflexes do not depend on the brain stem at all.
They are strictly a sensory nerve to spinal cord insertion and back to muscle.
Want to find out if there is no reflex left?
Strictly a Hollywood shot.
One of the resaons for checking reflex response during a neuro exam is to see if the reflex is present, but also if the CNS inhibits the reflex correctly.
Without the inhibition of the reflex loop by the brain, reflexes apppear hyper-active.
The patella reflex, for example, becomes a large jerk of the entire lower leg instead of the typical small reaction that the brain inhibits quickly. Any defect in the cord (or higher) that slows or prevents the inhibition indicates a possible problem.