First off, sorry it took so long for someone to answer your question:
My experience has been that the majority of students in these majors continue on the further their education. It's a common degree for pre-medicine, pre-physical therapy, pre-occupational therapy, pre-athletic training and pre-dietetic students. At my alma mater the vast majority of students (about 3 out of 4) studying exercise physiology continued on to graduate school. But not all continued on to medical professions: there are advanced degrees in sports conditioning, exercise physiology, or sports management that some of the students I worked with pursued.
If you do not want to pursue an advanced degree there are careers that you can complete with only a BS in exercise physiology (or strength and conditioning):
Strength and Conditioning Coach: work with HS or college team
Tactical Strength Coach: working with police, fireman, etc as a strength coach
Personal Trainer (although at least a minor in business/entrepuership is strongly recommended. It should also be noted that personal trainers are not required to have ANY college experience so this career would not pay a lot unless you open your own studio/gym/online company.)
Manager at a large gym such as Gold's gym or 24-Hour Fitness (However, it may be recommended to get a BS in business with a minor in exercise in that case.)
Exercise Physiologist: working in a hospital setting monitoring high-risk patients during exercises (essentially VERY skilled personal trainers. A basic description can be found at: https://explorehealthcareers.org/career/sports-medicine/exercise-physiologist/)
Physical Therapist Technician: (but if you earn an associate's degree in physical therapy you'll actually earn more as a therapist assistant.)
Hopefully that's a good start, you can learn more about a lot of these careers by visiting a few organizations that represent them. I personally have a lot of respect for the ACSM, NSCA, and APTA, and will link all three below.