This sentence is correct and natural as originally written in the CNN news story.
The phrase "after she saved", in this context, means something like "following her saving" or "as a result of her having saved". Both these phrases work with present perfect because they indicate the reason the dog is being hailed as a hero now. They don't mark the time when the dog was hailed as a hero.
As to your suggested change, if you mean "since" in the sense of "because she saved the man's life", then yes, "since" works. But if you mean "since" in the sense of "from the time she saved the man's life until now", then no, that's not the intended meaning. The phrase "after she saved..." does not indicate the time she was praised, only the cause.
Only the tense, present perfect, tells us the time of the praise.