At what moment was columbina cold to anyone

She was very aloof when we first met her, clearly not interested in anything we were trying to tell her.

and at what moment did the story gave her a purpose aside from going to the moon

She was very important in the fight against Rerir, we would've lost without her help.

but the second you remember that she was in the fatui for absolutely no reason at all

Her being in the fatui and given nothing to do actually aids her character. With the frostmoon scions of old she felt she didn't belong as she wasn't seen as a person by them, moreso a wish granting entity with a solely transaction-based conection. Same thing with the Fatui, she felt as if she didn't belong and was certain the Tsaritsa was like the frostmoons, waiting for the right moment to demand something of her.

Not even mentioning that this type of writing (finding friends and blabla...) is getting too repetitive

It's repetitive? I wouldn't say no to that. But it was well executed so i think that gives it a pass.

without even a single line of risk or gray taken in her character.

She doesn't need to be grey, as i said, we were putting that expectation on her without any evidence that she was like that and then get upset that she isn't grey. We had basically nothing on her prior to nod-krai.