In terms of optimisation analyses, to what extent has AD&D (2nd Edition)'s Charisma stat been analysed? How is it appropriate to proceed in an optimisation analysis of Charisma?
In terms of RPG history, the Charisma stat in 2e appears to be a bridge between rules that enable simulationist dungeon-crawling, and rules that enable character expression. 2e began with Charisma acting as a "pre-adventure" multiplier of action, in particular through expanding the party with NPCs. It also governs the reaction of non-Party NPCs and Party NPCs to encounters prior to the commencement of combat.
The 2e Charisma rules encourage henchmen and hireling spam, effectively enabling an "any-class" controller (with the Paladin being a specialist at this domain due to stat-requirements). The 2e Charisma rules also encourage non-combat encounter resolution through bargaining prior to combat, and imply the possibility of partial combat resolutions through parley, quarter, surrender and retreat (at full XP earnt IIRC).
A provisional analysis, to my mind, would indicate that making Charisma the centre-piece stat would allow for cheaper XP per unit of effort, possibly at the cost of treasure (and the remainder of the party being irritated at the controller). It would also change game dynamics from the "heroic four" to managing a pack of underlings. In general, it poses an optimisation problem as these "2e controller" powers resolve outside of the standard resolution systems for 2e (proficiency rolls, combat).
(Some interesting features here could be due to the system being insufficiently internally analysed, leaving exploits; or, this could be indicative of general "multiple resolution system" problem in optimisation).
Best Answer
The 2e Charisma Rules are direct continuation of the original rules in the 1974 edition of Dungeon & Dragons. Charisma was created to reflect the fact in history and myth there were individuals that were able to attract a band of followers to accompany them on their adventures. For example Jason and the Argonauts. That at some point during the development of D&D in either Arneson's Blackmoor campaign or Gygax's Greyhawk campaign this was pointed out. Afterwards guidelines were created to determine the exact number of loyal henchmen based on a character's charisma along with morale modifiers for hirelings.
Also a distinction needs to be to made between what D&D calls a henchmen and hirelings. Hirelings can be hired by any character with any charisma. The only impact charisma has is on loyal and morale rolls. The morale of hirelings dominate their interaction with PCs. Henchmen in contrast are considered members of the party and devoted to the PC they gave their loyalty too. Henchmen are usually only gained at higher levels particularly once a character reaches "name" level around 9th to 12th.
Most of the mechanics in 2nd edition and earlier were created to reflect some aspect of the fantasy genre and settings that Gygax and the other TSR designers worked with. D&D 2nd edition was noted for tailoring the base rules for different setting that were presented in great detail. Birthright, Dark Sun, Spell Jammer, along with the various historical sourcebooks. AD&D 1st and D&D were more of a melange of different ideas drawn from various works of fantasy.
The actual mechanics were not designed but rather evolved through actual play in the Blackmoor and Greyhawk campaigns and achieved a form recognizable to most gamers after the publication of the Greyhawk Supplement for Original D&D. Everything in D&D, AD&D 1st, AD&D 2nd, is a evolution of what was set down in OD&D + Greyhawk rather a holistic design effort like what occurred with 3.X, 4e and now D&D Next.
Because of how AD&D/D&D evolved there is little to stop abuse within the mechanics. Instead the game explicitly states that it up to the referee to apply his best judgment. That ultimately the rules are just a guideline to aid the referee in his adjudication of what the players do.
In the specific case there are several consequences that the referee can impose on players taking excessive numbers of hireling. The most important of which is that you need to feed and maintain their equipment. Another is the effect on the character's long term reputation as a result of the player's treatment of hireling. There is also the problem of moving large number of people through the city/countryside/wilderness and their potential to attract unwanted attention.
However on the flip side the referee should also be prepared to deal with highly charismatic characters whose players successfully attract and maintain large numbers of henchmen and hirelings. I can't stress enough that a good referee will be fair about his rulings.