I would say that the tipping point is when the industry leader went in for plot-based adventures, in which the main structure (the plot) cannot plausibly be replayed in the same campaign. By comparison, a site-based adventure's main structure (the site) can be replayed in the same campaign without straining disbelief. Just restock the dungeon, tip off the PCs that new evils have invaded, and away you go! (Plus, players who take good care of their old maps get rewarded for their diligence.)
On this measure of replayability, a "plot-based" adventure is replayable as well - just work out a new set of problems/circumstances for the NPCs to be caught up in, that relate in some way to the PCs. And the players who took good care of their notes on history and personalities of PCs even get rewarded for their diligence!
(I take "plot-based" scenarios in this context to be scenarios where social relations between NPCs, and between NPCs and PCs, are the dominant subject-matter of exploration, and where the payoff for play is something other than successful looting of a dungeon.)
The difference between "plot-based" and "site-based" adventures in my view is not primarily about railroading vs player freedom, nor replayability. "Site-based" adventures of the KotB variety seem to be primarily aimed at a type of exploration-heavy gamism. This sort of play is also emphasised by Gygax in the final few pages of the 1st ed PHB (Gygax's DMG also emphasises this sort of play, although in my view not quite as consistently, and in some ways more obliquely - eg the discussion of time in the campaign, many of the random tables and lists of traps and tricks, etc only make sense in the context of this sort of play). "Plot-based" scenarios are aimed at a different sort of play experience - I personally think the Dragonlance/2nd-ed style adventures are a failed attempt to deliver that experience, but that's a further point.
So why did it evolve into one character per player? Perhaps it was influenced by other RPGs. And perhaps it was someone just skimming over the henchmen and hireling rules. And by not understanding them
In some cases (I would consider myself as one such) the players in question did understand those rules but didn't particularly care for the play experience that it produces.
I don't know whether Moldvay Basic is considered new or old school. Or the sort of play described in the British paberback "What is Dungeons and Dragons?" (which was also available in Australia, published by Puffin - I don't know if this was available in the US). But these all assumed one player per PC, and in the examples of play had one player per PC, with the NPC henchmen being hired by the party rather than a single player, and clearly being a secondary participant in the adventure. (I think the example of play in Gygax's DMG also involved one player per PC.)
From at least the 1980s, if not earlier, it seems to me that there were a range of approaches to the game - some players liking the idea of role assumption and playing a PC through an adventure, but not particularly caring for all the other tropes of classic D&D play.