I also think Penelope Yin became a member of the Freedom Phalanx because she wanted to be one really Really REALLY bad. She's an untrained psychic that may be the most powerful one ever (maybe even stronger than Sister Psyche, so it's said)- then she's given an amplifier to boost her powers and help her control them better... yeah. It would also explain how the rest of the group seemed to make poor decisions without them realizing it (they were all being swayed by a young, eager, well-intentioned super-psychic).
Not to shoot down other people's headcanons, but let's look at the situation the Freedom Phalanx was in when they recruited Penny:
1. They just lost not one, but two of their founding members, their only incarnate, and their outright leader. Sister Psyche was surely not just close with Manticore, but all of the FP because she likely provided some level of psychological comfort to them as a psychic. Then there's the fact that Statesman was old enough to have been around when most of the FP were children, they grew up idolizing him, their parents idolized him, he probably seemed like an unchanging feature in an uncertain world. And then he was gone.
2. Around the time Statesman was killed, the Praetorian war was still in full swing, so the FP were dealing with the knowledge that their own counterpart to Tyrant was now missing, and the FP were having trouble fighting off the regular Praetors and even IDF foot soldiers who were getting jacked up on Incarnate BS powers from nowhere. Which brings me to point three...
3. Incarnates were popping up left and right now, and the FP was losing relevance with shifting paradigms even before Statesman's death, they likely felt people were losing faith in them and they were being left behind and becoming obsolete whereas Arachnos, while honestly only as strong as the FP had a greater overall staying power and ability to project itself because Arachnos is a national army of Etoile, Recluse isn't just himself, the four patrons, plus underlings. Recluse has arbiters, VTOL gunships, dozens of fortresses and hidden bases, super weapons, and more at his disposal. The FP was, when you think of it, a pretty humble super group. (Compared, to, say, even some player groups)
4. Psychic threats are extremely common in the City of universe, the FP had lost their prime psychic and
needed someone to pick up the old slack and Numina sure wasn't going to do it. So who did they decide to grab? The girl who was so strong when they needed a psychic to manipulate reality itself with the psychochronometron so the rules it operated on no longer worked they immediately overlooked Sister Psyche or anyone else, they looked to Penny.
5. Penny is powerful, insanely powerful, nigh incarnate levels of power as psychics go. And also young and untested. The Freedom Phalanx had probably opted to recruit her in hopes of mentoring her and, if nothing else, serve as a kill-switch if her psychic powers went novas like Sister Psyche's nearly did.
Addendeum: based on your headcanon sister psyche was mind riding in her husband's head, you can bet someone who wasn't a psychic themselves wouldn't be thinking too clear dealing with two minds in one head.
So, in summary: there's a lot of reasons the FP would be acting weird, edgy, and frustrated and why they would pick Penelope Yin as a member.
Plus, the FP was probably getting looked at by the Feds due to them having a long history of only Caucasian members and spending a majority of their time focused on dismantling an organization run by an ethnically diverse group.
I mean Arachnos itself originated as Italian fascists, Imad Malik, Ernesto Rodriguez, Belladonna Vetrano, Tamura Shirai, Irena Rudenko, and Paolo Marino with only Gideon Ray, Kelli Forston, and Stefan Richter seeming the odd ones out. Also, man it's weird to list those off after knowing them by their villain names. Just imagine how it would sound if the Arachnos inner circle addressed one another mostly by their real names.