One thing to keep in mind with a public health/health science major is that at some colleges these majors take the non-science major versions of bio, chem, physics etc. Only science major versions are acceptable for medical school admissions. If your college requires non-science major versions, you will need to check with the department to make sure substituting the science-major version is acceptable for fulfilling graduation requirements.
The only advantage is that a biochem major may have slightly better post-college employment prospects than a bio major.
There is a glut of unsuccessful med school hopefuls in the job market with majors in bio, chem or biochem.
You can literally major in just about anything and go to medical school. But you will have to make an effort to include the expected science coursework in your schedule. If you have room for a minor in your college schedule--you have room to complete med school pre-reqs. D1 & D2 had med school classmates with majors ranging from agriculture (forestry) to human geography to music composition to women's studies--plus all of the usual - biology, neuroscience, BME, biochemistry, chemistry. (FWIW, neither D had one of the "usual" pre-med majors.)
Find a major that appeals to you and that you can do well in since a strong GPA is very important for successful medical school application. If that major also improves your post-college employability prospects, that's a plus.
Every pre-med need a Plan B career.