Could it be me playing on Easier instead of Normal/Hard, then? For all my Akir Princess runs, ironically, the only times I was struggling was when I DID want to be Puppet Leader: as I mentioned before, Foraging on your own seems ridiculously exploitable* (thanks, Bedlam) so I hardly ever dropped to Alluring**, let alone Desired or Obsessed Over. I actually have to against the grain to get PL: rack up slutty traits, send that bimbo on missions she is sure to fail, get Strict Supervision and a dominant bedwarmer, fish for a slaver packed with dominant traits, and so on; it never happened "naturally" for me, I had to MAKE it happen, in other words, I never felt endangered unless I put myself in danger. Which, by the way, is something I look for in games: "you can un-fuck almost anything you fucked up at, and if you cannot - it was your choice, not a series of bad RNG rolls"; player agency, you know.
Unrelated, but there is one thing I cannot wrap my head around: being overthrown. From what I understand, going on a long assignment is a prerequisite (otherwise things just go the Puppet Leader way), but beyond that...I mean, I
have seen End of Day messages about it - "saved because they could not agree on who should be next leader"

- but what other factors are there? Say, worst case scenario: what traits/aspects combo would "make them agree" and establish a new leader? I get the hint is Golems like Ascendancy and Harmony/Malice, but which parts are the "mechanics" for getting overthrown, and which ones are just there for storytelling purpose of "Golems are scary beings; if you see them - RUN; if they see you - RIP".
*Even on Normal, Foraging shows ~70% Success and ~10% Crit, both scaled further by other variables like Slaver Training, Night Prep, Aspects (yeah, I always pick Unlimited Aspects), etc. I have around 5% chance of Failure and Disaster
combined. And succeeding often keeps Expertise up which translates to high EXP gain...which itself translates to frequent level-ups AND a level-up bonus of supplies, so I basically go "5 Supplies (without other bonuses) -> 5 Supplies -> 5 Supplies -> 5 Supplies -> 5 Supplies + 100 from Level-Up", and with a bunch of those buffs I now have something like "20 supplies + 300 from Level-Up" every week. I would have run out of Gold in the beginning without the Excess Supplies assignment.
**According to Forced Encounters FAQ in-game, "Respect not being at Alluring, Desired or Obsessed Over sets the variable to 0", and you need a value of 105 or 120 or something to even roll for a Forced Encounter, so unless I misunderstand "sets to 0", it is impossible to get PL without dropping to at least Alluring. And like I mentioned, my camp sucks so much (these are whatever random slavers I was dealt at the start) that my nothing-but-foraging Leader has the highest respect (Beloved) out of everyone there, others are anywhere between Liked and Desired from constant failures.