ATCs are way more suited to handle the variety of potential problems an athlete or team might have. I am both ATC and DPT. An ATC is a cross between an EMT, an orthopedic PT, and a PA, with some of the skills of each and all of the skills of none. I agree with Doc Wagner, a GOOD ATC is a team physician's best friend.
In PT school, we never talked about, let alone practiced an extraction from a pool of a diver with a cervical injury, or a gymnast with a cervical injury from a pit of foam blocks, or discussed in detail eating disorders with high school dancers, or how to treat a blister etc . . .
In undergrad athletic training, we didn't talk about cardiac rehab, or stroke rehab, or Parkinson's, or the potential consequences of CHF, how to deal with someone with dementia etc . . .
ALL professional teams have trainers, SOME (most) have PTs.