Ok, time for some history clarifications again. My understanding of the situation is that D&D was originally phrased as a inspiration booklet for minimal rule refered play (and as marketing material for chainmail, a more traditional rules based wargame). Actually putting structure to it making it reasemble an actual ruleset was as far as I known firstly done by enthusiasts by the D&D concept that also wanted more rules bound play.
The further profilation of rules addition and hardening was as far as I understand fueled by demands from the turnament scene, that needed a way to make the game play more similarly across DMs.
I hence think it is incorrect to credit Gygax and Arneson with recognising the value of structure. Rather I would say au contraire - they where fighting tooth and nails to keep the game as unstructured and free as posible while harsh market reality was forcing them to move the game further and further from that ideal.
Well, I played since the mid-'70s. I do not know Mssrs Gygax and Arneson, nor ever played with them. I did play TT wargames, and D&D. So, certainly Chainmail came first, and the 'fantasy supplement' formed one of several strands of development which combined in the form of early D&D. The 'Campaigns & Crusades Society' was a group who, as I understand it, built fantasy campaigns using Chainmail, the fantasy supplement, and some sort of campaign 'rules'. I expect it was all ad-hoc. Meanwhile Arneson was involved in the 'Banania' Braunstein, which is loosely based on ideas from FK and various other military/military-adjacent training techniques, as well as possibly some other existing RP and wargaming traditions (Wikipedia has a bunch of footnotes pointing at some different publications).
Dave Arneson and others then invented the idea of a location-based adventure using persistent characters (first used in a game called 'Brownstone Texas'). Dave mixed in rules from Chainmail, some mechanics stolen from another game he designed (a naval war game as I understand it) and then developed the basic ideas of classes, levels, and dungeon exploration as Blackmoor, which was located on the Castles & Crusades Society map of the "Great Kingdom" and environs (which became or at least got folded into Greyhawk). At some point the C&CS thing ended/died and Gary took up the task of further codifying D&D into a workable product. I don't think the object was to boost sales of Chainmail, as at that time Gary didn't own Chainmail, it was published by 'Gideon Games'. Anyway, Gary's first idea was to sell the idea to Avalon Hill, but they were not able to see the possibilities and turned it down. Thus TSR was born.
If you read the AD&D PHB and DMG front material you will quickly learn from Gary that codifying D&D was a significant concern of his. Not just in order to make tournament play feasible, but also in order to create a more salable and teachable product. This was the goal with Holmes Basic, which attempted to clean up and codify the existing D&D rules (at least a core part of them). This had nothing to do with Chainmail, which by that point (1976) was rapidly becoming a side note. In fact TSR published Swords & Spells around this time, a mass combat system specifically designed around the D&D "alternate combat system" which is the one Basic and AD&D adopted, dropping all reference to Chainmail, along with the original usage of its combat system (which was originally the main system for combat).
Thus I find no evidence that your conclusion is correct. It may be that Dave and/or Gary were interested in limiting the scope of the rules, but we don't know that. We know next to nothing about what Dave thought. As for Gary, he went on to write UA and AD&D added a HUGE number of new spells, for instance, which seems clearly not a minimalist approach, nor are the large number of rules in the DMG very compatible with that idea. I'm OK with the idea that Gary believed in a loose toolkit of subsystems that could be employed, not employed, or hacked as needed by the DM, but I can't see how this is in any way compatible with the FKR idea of 'invisible rulebooks'.