I 100% agree with what has been said already, cost is the big issue. If it doesn't make sense financially but you really want to do get the credentials you can look into a sport residency. Cost might still be an issue as residency isn't exactly free, but it would likely make more financial sense than the DPT/ATC combo due to what truthseeker mentioned.
You will also have to consider what your career goals are. A lot of potential PTs get into the field because they want to work with athletes but PT is so much more than that. As mentioned above, orthopedics is only one area of practice for PTs. The scope is pretty big when compared to an ATC, so in that regard PT is more flexible, better hours, usually better pay, but working exclusively with athletes is rare. Heck I know a clinic who is the exclusive provider for an NBA team and it only makes up 5-10% of their case load at any given time. They still treat mostly non-athletic injuries for the bulk of their day. Most ATCs only work with athletes and it can be exhausting. Of course not all ATCs work with athletes, I met a gentlemen a few months ago who now works in an industrial setting as a ATC and swears he will never go back to athletics.