Chani was not only a sayyadina for the sietch, but she was one for a sietch with an old and dying Reverend Mother. We have no reason to believe that the Fremen RM's conducted true BG training among the Fremen - even among the sayyadinas. We pretty much know they didn't since no Fremen knew the BG muscle and nerve control or the BG combat techniques. And we know that the method the Fremen RM's used for conducting the agony for new RM's wasn't the same as that used by the BG themselves. Nevertheless we know that new RM's were made from amongst the Fremen, and that those who were sayyadina would be the ones selected. Shall we say we can safely conclude that Chani was in line to be the next RM for sietch Tabr, or maybe even the official replacement were it not for Jessica's arrival?
The fact that Chani was Liet's daughter (and Stilgar's niece, by his sister) not only would mark her as being a prominent figure in Fremen society, but would also mean that she would have been highly trained in various ways. She was certainly trained in the 'ways of the Fremen' as a sayyadina, even though we don't know what specific training Ramallo might have given her to help her pass the agony. We know she was a good fighter, and even sly enough to sneak up on Paul in the rocks unnoticed. We assume that Liet would have trained her in the ways of planetology just as Pardot had trained Liet. We also know that she would have had experience observing Stilgar rule and probably had his wisdom to draw on in her learnings as well.
With all this going for her, the moment Paul and Jessica swooped in we don't hear a peep from her, not a complaint. Jessica takes the role Chani was most likely groomed for - to be the next RM for Tabr - and Chani says nothing since it's for the good of the sietch. Paul takes his place by Stilgar's side helping run the tribe, while Chani is simply 'his woman' - and still no complaint. Even Harah, who was by any standards a good woman and a realistic thinker, wasn't as calm in the face of all this as Chani was. Chani even recognized right away the greatness in Paul and was attracted to it, rather than fearing or resenting it.
All of these things even in the young 14 year old Chani mark her as a remarkably balanced and impressive girl. I could go on about her patience during Paul's marriage with Irulan later on, about her loyalty to Paul without it going to her head that she was the consort to a prophet and mahdi, and about the fact that she was always more concerned about Paul's well-being than about herself, but I don't even need these other things to realize how amazing she was.
And let's not forget, as others point out, that Other Memory is not actual people; it is just memories. Chani's memories may have been tempting for Ghani to bring up, and Chani's love may have been powerful enough that Ghani couldn't stop herself dwelling in it, but this was NOT Chani doing anything. It was all in Ghani's mind and imagination, struggling to make sense of her own memories of other lives.