It sounds like you guys don't play multiple turn-based games of Reversi at once.
"...it's nice to know as player and have that experience..." It's an option players can choose to enable, so won't affect anything.
"...a 3-second delay is not going to help if it's a turn-based game." It's a 2-player game. After you make a move, if you wish, you can wait x seconds and see the other player's only move made automatically, then it's your turn again, of course it will help. Your point is only valid in games with more than 2 players and even then, the advantage is deemed to far outweigh this disadvantage in games such as Nidavellir.
"...there's always (well, not always, but you get my point) another possible move: resignation." Ha, good point and technically, true, though of course advantages of automation far outweigh this disadvantage, even if it is one.
"But does having only one move always mean you don't win? What if it was literally the last piece (1 square left) and you are winning? I know Trick of the Rails has options for last card too. What if you put an auto last move then XD" In many games of Reversi it's possible for one player to have only one valid move, usually towards the end of games and usually losing players, but absolutely not exclusively so in either case.
"You really need to soak up that humiliating feeling of being forced to play such a bad move, that's the way you'll learn not to get in this position again next game." Disagree 100% and could even put players off from advancing. The main advantage of automations far outweigh this disadvantage even if it is one.
"What if you're really short on time, and having the opponent making a turn gives you some extra thinking time." This could be used in argument with Nidavellir, King of Tokyo and many more too. The unlikelihood of this happening in turn-based games makes this disadvantage miniscule in comparison to the advantage gained.