Almost all of the additional endgame rules beyond the “more than half” thing were built out of old tournament rules and rulings, where they needed more defined ways on how to handle games with multiple players, play styles that didn’t focus on getting the most gold, and all sorts of other odd edge cases that came up.
Ending the game when enough players had been wiped out was added so there wouldn’t be situations where some players who were technically still part of a game would be stuck waiting for the others to finish collecting treasure.
Ending it quickly (the next turn) after enough players were out was intended to address situations where one player would ignore treasure entirely and would instead build a fleet designed solely for attacking (in the days before action limits of any kind, often it was often just a single, massive, multiple-action warship like the Titan or Harbinger), under the originally-legal assumption that all gold in play would become theirs by default when there was no opposition left. Other players without a similar setup wouldn’t stand a chance, and when both players were set up for that, games could come down to a matter of who went first or who risked leaving their home island first, so that ruling and other similar rulings were created to dissuade those types of tactics, at least within “normal” games.
The printed rulebooks never really kept up partly because they were intentionally kept simpler for the more casual players who might pick up a few packs to try out the game. Also, and maybe even more significantly, there always seemed to be a large disconnect between whoever was editing the rulebook, the designers, and the tournament rules. That issue continued right up to the very end – just look at how long it took to finally get the action limit included in the actual rulebook.
As far as the “immediately” versus “start of turn” issue, as far as I know the official timing has always been the start of a turn. I can see where the distinction would matter, in a scenario where a player had ships in scoring position but used one of their first actions to knock out an opponent’s last ship. Delaying the endgame until the next player’s turn means they don’t have to be super careful about the precise timing of when they make that last attack, and it also covers an edge case where the “eliminated” opponent might still have a facedown Shipwright or a recruiter that could bring one in.