Re-read that chapter and didn't see anything about a misteryous fragment or a saint quest, the job change is activated from praying when kneeling, MC spoke aloud so what she said could be a specific keyword to activate it too, the goddess noticed the bracelet and skipped on a "tribute", so is likely that for any class change without the bracelet it would require a exchange, maybe that is a novel thing?
The goddess even thought of not having anymore believers, so without the prayer being made in a specific way with the specific words it looks like the believer can't be connected, messed up for her and her followers, but it makes almost impossible for most people to activate it and the bitch have even less chances thanks to it as the first step of the process is "below her".
i couldn't find what you said in the first 3 chapters, maybe it was mentioned later and I forgot through the months, a re-read is a must.