A couple things that hopefully add insight:
One: Note that this comment is about "any single campaign" (with more verbiage in that regard in the answer to which you've linked). Those players may or may not be a single game session, i.e., all at the table at one time. My understanding of Gygax's early games is that he had an "open sandbox" style in which he was running a game every night of the week and a variety of different people showed up any given night.
Two: The original D&D rules are so stripped-down compared to modern rules that there's almost no debate/deliberation/book lookups needed to run the game. In this regard, you can give a new player a fighter and teach them no rules whatsoever, and they can perform admirably with just common-sense and raw role-playing the situation and real-life resources. This is also combined with the expectation that the DM will be making reasonable adjudications, and the players will not be rule-debating them, at all times. Therefore the play is much speedier: an average action of 1 minute per player would be considered painfully slow.
Three: Initiative is for the team, not the individual. So there's no overhead in tracking or evaluating who's up next; when it's the players' turn in combat we can just go around the table in order of seating very quickly (like other more basic boardgames). Other rules are likewise geared towards the party and not the individual.
Four: Games are expected to use a position referred to as the "Caller" or team leader. In particular, all of the exploration decisions can be relayed to the DM by this one senior player, which vastly cuts down on the communication time. From original D&D (1974), Vol. 3, Example of the Referee Moderating a Dungeon Expedition:
REF: Steps down to the east.
CAL: We're going down.
REF: 10', 20', 30' — a 10' square landing — steps down to the north
and curving down southeast.
CAL: Take those to the southeast.
REF: 10', and the steps curve more to the south; 20'. Steps end, and
you are on a 10' wide passage which runs east, southeast, and west.
There is a door to your left across the passage on a northwest wall.
CAL: Listen at the door — three of us...
In this entire section, there is no instance of a player other than the Caller (CAL) communicating with the Referee (REF). That dramatically cuts down on play time. (Note that this has the added side-benefit that splitting the party due to a disagreement is nearly impossible.) This is elaborated on in the later Advanced D&D Player's Handbook (1978), in sections on Obedience, Organization, and Successful Adventures; and the AD&D Dungeon Master's Guide (1979), with a longer example of play in which again the DM is almost solely interacting with a leader character (LC).
Personally, I still run games using original D&D rules, and they can be very quick and fast-paced in a way that later games ultimately frustrated me. I've had up to 12 players at my table and it worked very well; I feel like the more players the better and more exciting the games are (and I have players asking for more sessions of this game at the mini-convention my friends run every year). I do still use a Caller, but it is a very low-key usage; the Caller has a responsibility to manage party deliberations and present consensus to the DM, rarely making a personal call except if there is no agreement and we need to keep the game moving (more on my personal blog here).
Late-Era Gygax Reflection: I found more information on this point from Gygax in the ENWorld Q&A thread (dated 14th February, 2005). It seems like the most he dealt with at a single table at once was around 20:
For about six months the typical number of players in an adventure
session in my basement was 18-22 persons packed in. That was when I
asked Rob Kuntz to serve as my co-DM. Getting marching order was very
important. Of course most activity was dungeon crawling, so actions
were just done in order around the table. Be ready or lose your
chance! Stick with the party or else something very nasty is likely to
befall your character away from the group. The sessions were fun but
somewhat chaotic, lacked most roleplay, and surely didn't allow for a
lot of one-on-one time player and DM.
I DMed a con tournament with 100 entrants, and i managed 20 in each
group. I took time to check individual actions there, as it was an
outdoor adventure. Each session ran four hours, and a bit. I was
surely tired when that was concluded, but to the best of my knowledge
all the participants had a good time of it, even those on the teams
that didn't finish in the top spot.
Personal Experience in 2018: At GaryCon X I had the opportunity to play in games on two nights run by Bill Webb (Necromancer Games, Frog God Games) that had 20+ players at a single large conference table (27 Friday night and 23 Saturday night, as shown below). Lightweight Swords & Wizardry rules were ostensibly in use. The DM never looked at any rules or notes while the game was in progress; and only ever used a single d6 and d20. PCs were mostly 1st-level, with a few legacy PCs of 2nd-6th level, and no one had any ability for more than one action per round. As above, team initiative was used and actions were taken around the table in seating order; the caller was not used.
Much of the action of the game was the DM allowing the players to observe a monster, and then discuss plans/strategy almost without limit until combat was engaged. No wandering monsters were in use, and also nothing one could parley with (monsters were undead, aboleth, purple worm, displacer beasts, etc.). A tapestry-sized illustration of the dungeon was on display, so that mapping and navigation were non-issues (although we found several secret chambers during play; of course, this is very different that what we know of Gygax's games). General process was for DM to possibly leave the room and let players discuss strategy for 20-30 minutes, then engage in combat for 20-30 minutes, then proceed to next encounter area. Very interesting experience; there's more description and photos at my friend Paul's Game Blog.