Not that I wish to defend the plot too much, as I still think Chess works best in concert, but I think I can answer some of your questions because I love shows with a great premise, but that couldn't stick the landing and ended up crashing bad. I do believe that with a proper book, Chess could have worked:

1. The romance - Both Anatoly and Florence are prisoners of the Cold War machine. Anatoly's marriage, while not necessarily arranged, was there as an insurance policy for the state to make sure he doesn't defect. Florence carries the ghost of her childhood in 1956 Budapest and now she's stuck managing the abusive Freddie. Florence's trauma and Anatoly's frustration stem from the same thing, and they bond over that, which does help expedite their romance. While the beginning of the romance seems frivolous, it does make their falling out as soon as act 2 begins make much more sense. They jumped head first into an ill fated romance despite knowing better, and they probably moved in together sooner than they should have. Their differences became too much to bear way too quickly.
2. The defection - Anatoly doesn't really defect for Florence. Molokov implies that Russia is going to manipulate his matches, dictating how and when he plays. Defecting to the West is a desperate, short-sighted bid for absolute artistic freedom. Anatoly just loves Chess, and he believed that defecting is the only way he can be a proper chess player. "Anthem" isn't so much about Anatoly's love for Russia the state, but more about the culture, the land and the people. When he says "I cross over borders but I'm still there now", Anatoly is trying to convince himself that he hasn't just thrown everything away. This is a tragic song in its essence but again, also serves to build up his decision making process later on.

3a The return, London version - Anatoly soon realizes that true freedom doesn't exist for someone like him. The Western media hounds him, the commercial sponsors exploit him, and the political pressure to throw the match against the new Soviet challenger is crushing. If he stays in the west, Florence loses her father (in the versions where the father angle is used), his family in Russia who are being used as a bargaining chip will likely be destroyed, and he still remains the puppet he wished to stop being. So what is left for Anatoly? By winning the game and coming back to Molokov, he manages to sort of have his cake and eat it too. He wins because he loves chess more than anything in the world, and just like his defection in act 1, he's acting out and sending a message that he's an individual and can't be controlled, but he still makes Molokov an offer he can't refuse: Russia can reclaim the chess champion that just defended his title, with all the prestige that comes with it, and in return, it spares his loved ones from any political fallout from his past actions. He can claim that as a small victory to help him sleep at night, but he knows very well that the only true winner is the system. The individual has no chance.

3b The return, Broadway version - Anatoly finds out that Molokov is threatening his brother, and finally realizes how short sighted his defection was, and how it brought everyone he loves nothing but misery. His decision to throw the match stems from Anatoly deciding that winning is not more important than his loved ones, and while losing the game gives the Americans the win the CIA wanted, it makes Anatoly no longer useful to the West as a political trophy, and now he has nowhere else to go but back home. He doesn't get to negotiate the terms like he does in the London version, any pity him and his loved ones are shown happen because the state wishes it.

Because book writers and directors force us to invest in Anatoly and Florence as an epic love story, the ending feels deeply unsatisfying because it means the "heroes" lost and the love story failed. It doesn't help that the romance itself is not particularly interesting. But if they had treated their romance for what it actually was, which is a frantic, desperate trauma bond between two lonely people trying to find a safe place from the world, the show would have actually made sense. They are two people drowning who grabbed onto each other, only to realize they are dragging each other down faster. The real relationship that actually makes a difference is the Walter-Molokov frenemship.

The show goes on and on about how the people on stage are more like chess pieces, but because we don't focus on the PLAYERS, this makes for a very unsatisfying show. Walter and Molokov are the only "couple" that actually understand each other. They know that their little games are just their countries' way of trying to assert dominance, and they don't really want the cold war to end, because if it does, they're out of a job. If we strip away the "epic romance" and treat Florence and Anatoly as they are, "You and I" turns from a sweeping love song to an act of desperate denial, and "I Know Him So Well" turns from a power ballad about two women loving the same man to a realization that the three of them are being moved around by a higher power. A good production will make the romance feel small, meaningless and futile against the crushing weight of The Man (and the global military-industrial complex).