Every notable tribe of people that has an identity has its legends, now not all of them are true yes but those legends are close to heart of those people.
Jauhar and Saka is a proud Rajput tradition, even those who have converted to Islam speak of it with great honor. If the Jauhar of first siege of Chittor is not documented by Muslim historians then that does not mean it did not happen. Later Jauhars were documented, so the tradition existed. Persian historians documented Jauhar of Ranthombore at the time of Alaudin, so the tradition existed even at his time so just because you can't find a Muslim source to confirm it does not mean the story was fake.