You're right that MPH admissions do not require (or even necessarily prefer) a BS in public health. However, majoring in public health might be a great choice if you're sure you want to pursue an MPH because it may give you a lot more knowledge about the field. It might also help you decide which subfield of public health to pursue (e.g epidemiology vs. health policy vs. behavioral science).
Would a major in public health also give you more opportunities to get involved in public health related internships or work experience? This is something to consider.
That being said, if you're really interested in say, psychology, there's no reason you can't major in that instead. I majored in music, and now I'm starting an MPH in epidemiology. So you really can't go wrong. (Just make sure to take some basic science and math electives if you major in something totally unrelated like music.)
ETA: A BS in Public Health is not really useful on its own. If you want to be a public health professional you will really need a master's degree.