The caller is an archaic role that is only relevant when the play group is very large. And by very large, I don't mean six or eight players, I mean ten or sixteen. Our sense of what a "large" group is has adjusted drastically downward since BD&D was written, and consequently the purpose and utility of a caller is no longer obvious.
The gameplay advantages of having a caller were all of the above. In early D&D it wasn't uncommon to have a dozen or more players in one gaming session. The role of the caller was primarily a table role: they were the liaison between the giant mob of players and the DM. The role of the caller allowed the responsibilities of table management, player organisation, maintaining speaking courtesies and discipline, and consensus-building in terms of course of action, to be devolved from the DM's chair onto one of the players. The chaos of having "What do you do?" answered with a babble of a dozen or fifteen voices all talking over each other is instead replaced by a single speaker for the player-group.
Of course, that still meant that the caller had to deal with that babble, but the division of labour meant that they were better able to do it than the DM. Also, it gave the players a more-or-less subtle motivation to be more organised and regimented in their communication—if shouting the loudest doesn't allow you to dominate the DM's attention because your declarations still have to go through the caller, and all it does it interfere with the caller's ability to bring a group consensus to the DM in a timely fasion (which may be deadly at times!), then you're much less likely to resort to impotent shouting and more likely to develop keen group decision-making skills out of necessity.
A far secondary purpose of the caller role was to speak for the group to NPCs, not just the DM. This is a natural extension of being the conduit of decisions between the players and DM, as the caller would speak to NPCs according to the plans of the group. But, as the speaker, their Charisma influences the NPC's reaction rolls, hence why the advice to have the player-role of caller be held by a player whose PC also has at least a decent Charisma. It was entirely possible for the caller to not be the in-game "face" of the party by delegating the role: where instead of saying "the thief will check for traps", the caller might say, "our charismatic wizard will negotiate with the mayor," at which point the direct communication would switch from between DM and caller to between DM and the indicated speaker.
This is not to say that the caller was the only means of communicating with the DM. It was a formalism to set default expectations and to alter the behaviour of the group as a whole when it was very large. But DMs could and would address individual players if that made more sense, such as when an NPC was directly speaking to their PC instead of to the caller's PC, or when the player's PC was separated from the group or otherwise perceiving things unique to their PC. The DM would also consult with the mapper (another player role with a mixed-in character role), time-keeper, and other distributed-responsibility player-roles, if they were being used by the group.
In smaller groups, having a caller was obviously not as useful, and was often dispensed with. In a game of BD&D today, even what we would now consider a "large" group would probably be small enough to not see any utility from nominating a caller. If you only have six or seven people around the table, the group is able to self-organise easily enough without needing a player-role leader, and the DM can gather everyone's desired courses of action and hold them in memory more-or-less well enough that the filter of a caller wouldn't speed things up much at all; so the loss of direct back-and-forth is less worth accepting if there's little or no gain.
