Can people with sleep problems benefit from a mental health practitioner?

Most certainly yes! Solving sleep problems is a common adventure in our society. Over the counter sleep aids are everywhere, with varying success and often with problems of their own. Melatonin seems to be most helpful and side effects low. If that's not helpful you can start with either your primary care provider or a mental health professional. More difficult cases may require sleep apnea or psychiatric consults.
Depending on cause. Many sleep problems can be symptoms of illness, but sleep problems can also be their own issue. For that a sleep doctor will be needed to clarify the diagnosis.