But how close does she really get, and how close do they remain?
For a time, and then she pretty much threw her over, taking out her frustrations on Siuan who did the best she could with incomplete information, and using her judgment which in terms of political calculation had largely been superior to Egwene's. Not to mention her experience, which is not a thing that ever compares well in Egwene's favor. And by that time Sanderson was writing them, so we can't get a good sense of their relationship.
That's not automatically explained by friendship. In fact there is no indication that they were ever even as close as either was with Siuan. The visits in the Tower were because she had a responsibility to Leane, to which her exhaustion and time constraints simply do not matter in comparison. I don't care if she's got broken limbs. Short of debilitating or life-threatening injuries, she OWES her captive subordinate that much, in a situation where Leane was imprisoned following her orders, and is not in a position to receive visits from anyone else on their side. That's not a matter of affection or choice, that's an absolute obligation. If Egwene thinks it was duty that put her on the boat (it wasn't), there is no way she overlooks her obligations to Leane. I don't think her time among the Aiel had such an effect on her that she took ji'e'toh at all to heart, but those who do so believe cannot admit that it is even possible she does not see the force of her duty in the case of Leane.

And on the personal feelings issue, Leane is also the only sympathetic individual to whom she has physical access. She needs the visits as much as Leane does.

Egwene would be visiting Leane every day even if they hated each other. If it was Faolain in the cell instead, she'd be doing the same thing. It's not an indicator of friendship.

See, I don't like this stuff. I don't like people suggesting Rand & Lan or Thom or Bashere have a filial/paternal relationship. I wouldn't like to see Mat adopting anyone as his first-sister (or Egwene for that matter), because they have actual relatives. It seems profoundly disrespectful in my mind to put another person in terms of a real relationship, especially on such short notice. And the relationship is pretty much typically one-sided, as discussed in other posts in the discussion. Egwene gets unique training, based on lies, and only starts paying the Wise Ones back after she is apart from them. And Amys was not a part of that, because Egwene ended their teaching relationship by breaking her word to Amys.

But I said, she is very good at making people like her, and having good times together. But that's a very shallow perception of real friendship. I don't like most of my coworkers, despise some of them and hold most in contempt, but we can still get along and joke together when we have no choice but to be together. We are not friends, but we can act friendly. And notice, Egwene does not exactly seek out their company. When the group that came through Jangai Pass has pretty much dispersed in "Lord of Chaos", Egwene spends a lot of time on her own, rather than chilling with the Wise Ones.

An interesting thing is how all the friends you posit here are at least a generation older than Egwene. They are people who have held or do hold high status positions. Someone whose dating history looked like this would be called a gold digger or social climber. The criteria by which Egwene seeks bonds of affection might simply be coincidental, but they are pretty universal. And too, these older, higher-ranking friends are developed as such largely at the expense of extant friendship with her peers, down to the point of outright neglect.