Either the transfer program you were speaking with didn't know what they were talking about, or you misunderstood them. First off, do you live in the USA? If not, then we're probably not the people to ask, as most of us are in the USA and are familiar with US requirements for this sort of thing. If so, then we can try to help you.
Yes, to work with athletes you need to understand kinesiology and will need to take a lot of kinesiology courses, however you do not need to major in kinesiology. There are several ways to do this:
1) Get an undergraduate degree (Bachelors degree) in Athletic Training from a school that has a certified program. Upon graduation, you'll take a national exam, and then providing you pass, you'll be able to work with athletes in high schools and colleges by providing coverage at their sporting events (basically, if someone gets injured, you'll be one of the first responders to attend to the injury, as well as helping out with preventive measures like pre-game taping, some hands on work, etc.). You may also work in a training room, where you help condition/train athletes. Some ATs also work out of PT clinics, so they will help out with patient's rehab (sometimes athletes, sometimes other patients who are in the clinic).
2) Get an undergraduate degree in kinesiology, or some other movement science, or really anything as long as you meet the prerequisites of the program, and then get a Master's Degree in athletic training. You'll need a master's degree (most likely) to work with professional athletes, or to be the head trainer at universities. However, if you get an undergraduate degree in something other than athletic training, and then decide not to go to a Master's program, you cannot become a certified athletic trainer. No matter how many kinesiology/anatomy/etc classes you've had, the only way to become an athletic trainer is to go through a certified athletic training undergraduate or graduate program.
3) Get a personal training certification (some other members have mentioned specific ones) that focuses in sports medicine-releated areas...this lets you work as a personal trainer to anyone who wants personal training, but if you get certain certifications you can be more attractive as a trainer to athletes. This just requires passing a test, and usually a high school diploma (some certifications require a college degree, but it doesn't have to be in kinesiology or any specific major...just the degree).
4) Major in whatever you want (many choose kinesiology, or biology, or something similar) as long as you complete the prerequisite courses, and then go to physical therapy school (a 3-year doctoral program, which is what most of us on this forum are doing/trying to do). As a PT, you'll work with patients who are injured and need to be rehabbed. Many PTs specialize in athletics/sports medicine, so they work in outpatient clinics and often see high school athletes, college athletes (if they are in a town with a nearby college) and occasionally professional athletes depending on the clinic. Some PTs may also be lucky enough to work with professional teams as the main PT when a player gets injured, but this is really luck of the draw as a PT...not many will get to do this (realistically). As a PT, you'll have to be open to treating everyone, not just athletes, because even if you work in an outpatient orthopedic/sports medicine clinic, you'll have patients of all kinds come in with those types of injuries - not just soccer player tear their ACL, so you have to be willing to work on the 65 year old lady who tore her ACL on her ski trip. It is possible to be a PT who primarily works with athletes, but it's hard to be that narrow in focus as a PT.
5) You said you didn't want to go to medical school, so I wouldn't suggest this option, but you could become a sports medicine physician (they mainly deal with sports injuries, but don't usually do surgeries...they'll also work with high schools and do sports physicals for athletes, etc.) Other options would be an orthopedic surgeon (so you'd be cutting into athletes, as well as other patients), or becoming a physical medicine & rehabilitation (PM&R) physician. All of these require medical school though, and if you're not 100% dedicated to it, I wouldn't go this route.
Honestly, I wouldn't get a degree in kinesiology unless you know you will go to PT school or a master's program for AT, because on it's own it isn't necessarily the most useful degree.
If you know you really only want to work with athletes, I'd highly suggest getting an undergraduate athletic training degree. That way, you can become licensed right after college and start working with athletic programs, and if you decide you want to get the master's degree or to go to PT school later on, you still can.
I hope this helps?