My only extended bird experience is with feral Rock Doves (a.k.a. pigeons), and wild American Crows, but from observing other wild species, I'm pretty sure that given the right temperament, birds are pretty damn smart and can be taught anything. The human 5000+ years associating with the Rock Dove has given us insight how to train them. First off, the racers and homers are not just returning to where they come from. They have an incentive: their mate. I think that any bird that leads a monogamous life-style would be far easier to train because you don't have to provide the incentive; they do that for you. Also, flock birds prefer to be around their own flock. Rock Doves are extremely social, though it can vary by the individual.
While a Mourning Dove is also monogamous, they are not flock oriented, and not as social as the Rock Dove. However, if there were another candidate for homing, the Mourning Dove would be on the top of the list. Another factor for homing or racing is speed, and the Mourning Dove has that as well.
On a 103 degree day, in August of 2018, I was walking from work to the gym, and I spotted a red-tailed hawk sitting on the roof of a single story house. I wouldn't have remembered it, but moments later a low-flying Mourning Dove passed over my head, just slightly higher than the roof of the house. At first I assumed that it must be injured because I hadn't ever seen a bird flying at half-pace.
Many of the larger birds don't need to flap much once they're aloft, but pigeons and doves have to keep their wings moving.
This entire story happened in a matter of a few seconds, but it's impossible to relate all of the details in writing in that short of a period of time. I stopped, turned and kept watching the Dove. Out of the corner of my eye, I saw the hawk lift off. Apparently it had the same idea that I did, and he decided that it would be an easy meal.
Now, the moment the hawk was in the air and on the chase, the Mourning Dove changed its wing beat to normal speed - and the hawk landed on the first utility pole that it came upon. And that was that. Not worth the effort, because they are too fast.