Dawn was more of a contest battler than a battler in the same category as Cynthia and Ash, so she had little use in a region that has no Pokemon Contests (I mean, did ASH go to regions that only had pokemon contests and not Gyms and the like?). And besides which, last I checked, the only female character who actually cared about honing her battling skills (ie, gym and league-related battles) in actual trainer battles besides Iris was Misty, so the third reason doesn't actually apply to her at all.
Then again, it's also likely that Dawn was only added into the opening as a way to promote her appearance and yet still have her leave after a few episodes. They had May and Dawn appear in one of the DP openings as well, and May was only around for a few episodes, and Misty, aside from her lure, never appeared at all.
And Dawn has to have been pretty quick to finish both Hoenn and Kanto before heading to Unova. After all, she left for Hoenn in that special to do contests in the region, and we already know that Kanto has contests as well, so unless she won Hoenn offscreen, I find it hard to believe that she'd enter a region that has absolutely no contests. It's unlikely that she'd interrupt her own journey just to visit Ash. Ash never interrupted his own journey just to visit his own friends, and even with May, when she went to Sinnoh, it wasn't to visit friends so much as get an inter-regional ribbon for a Pokemon Contest.