You are still repeating the same old fallacy: Confusing Joining and Recruiting.
Carrot WON'T be recruited, she will JOIN the crew. Meaning that her "arc", will only happen AFTER she joins (between the end of wano, and the end of the story)
You are looking at Carrot's story completely in the wrong way. What Carrot had, is a setup to a joining, not a recruitment, so OF COURSE she didn't had a save by Luffy, that's not the point right now.
Just like Robin and Nami's story came AFTER they joined, Carrot's one will come AFTER.
That's something Yamato won't have, why ? Because on the contreary to Carrot, yamato had a lot of screen time (too much actually) to have another later. Carrot don't have that weakness, Oda has perfect freedom to create an entire storyline for Carrot.