The book The KGB Plays Chess by Boris Gulko, Yuri Felshtinsky, Vladimir Popov, and Viktor Korchnoi, alleges some pretty wild measures that the Soviet government took to ensure that Anatoly Karpov remained the World Chess Champion.
For background: Korchnoi challenged Karpov for the World Chess Championship in 1978 and 1981. Gulko is a former Soviet chess player and chess writer. Vladimir Popov is a retired KGB polkovnik. Felshtinsky is a historian of 20th and 21st century Russia. Gulko is the primary author.
It alleges (among other things):
That Anatoly Karpov was himself a KGB agent (codename: Raul).
That the KGB had a contingency plan to poison Karpov's challenger, Viktor Korchnoi, in 1978 should it appear that Korchnoi was going to win his match against Karpov. (Korchnoi was a Soviet defector, and supposedly the concept of allowing a defector to win the World Championship was unacceptable). Korchnoi lost the match, and so this plan was never carried out.
That Karpov was strongly favored by the Soviet chess establishment in part because he was one of the few top chess players at the time who was not Jewish.
That Kasparov was under constant surveillance by the KGB in the early 1980s, who concluded that he had liberal, democratic, and humanist political leanings that were unacceptable to the Soviet government. This supposedly led to the Communist Party and Soviet Chess Federation attempting to prevent Kasparov from becoming World Chess Champion.
That the President of the World Chess Federation (FIDE), Florencio Campomanes, was a Soviet intelligence asset who was on the KGB's payroll (along with International Olympic Committee President Juan Antonio Samaranch). And that Campomanes called off the Karpov-Kasparov match in 1984 so as to prevent Karpov from losing the match (after he had just lost two games in a row).
That, while Karpov was entitled to an automatic rematch with Kasparov under the rules in place while he was World Chess Champion, FIDE changed the rules while Kasparov was World Champion to remove the automatic rematch, with the explicit intention to prevent Kasparov from winning any subsequent rematch should he lose.
I'm rather skeptical of a lot of these claims, especially given that both Gulko and Korchnoi are Soviet defectors with an axe to grind. I know Kasparov has claimed that the Soviet chess establishment tried to prevent him from becoming World Champion, but I've never heard him explain why.
Is any of this stuff true? This book seems to be the only pertinent source I can find, and I'm rather skeptical about a lot of this information unless there are other sources confirming this.
I don't know enough about the KGB and the Soviet government to critique The KGB Plays Chess (though I harbor some of your same concerns about the authors' motives). However, I can address an aside toward the end of your post:
I know Kasparov has claimed that the Soviet chess establishment tried to prevent him from becoming World Champion, but I've never heard him explain why.
In short, he didn't correspond to their image of the ideal Soviet. Consider Rashid Nezhmetdinov (1912-1974), a brilliant Russian chess player whom the government only allowed to compete internationally once (Bucharest, 1954). Why did the Soviet regime refuse to support his career? In large part, this owed to his Tatar ethnicity and birth in modern-day Kazakhstan. He wasn't Russian enough.
Now consider Garry Kasparov, born Garik Weinstein, in Azerbaijan. Garik's father was Jewish, and his mother was Armenian. After Garry showed significant potential, including an invitation to study in Michael Botvinnik's school, he changed his last name from Weinstein to Kasparov (his mother's last name is Kasparova). In his 1987 autobiography Child of Change, Kasparov notes, "People are puzzled that I changed my name from Weinstein to Kasparov around the age of eleven and assume this was to disguise my Jewish background. This is quite wrong. In any event, taking an Armenian name was simply exchanging one minority from another. It made no difference politically." (24). However, in his 2011 autobiography/collection of chess games Garry Kasparov on Garry Kasparov, Volume I (1973-1995), Kasparov contradicts himself by noting his family had a "family council of the Weinsteins and Kasparovs" in August, 1975, after his trainer, Alexander Nikitin, had insisted on the change. Kasparov quotes Nikitin extensively: "A year before that, with Botvinnik's agreement, I began insistently trying to convince [Kasparov's mother] of the need to change her son's surname. I had no doubts about the boy's brilliant chess future. From my work in the USSR Sports Committee, I knew what inexplicable problems, not at all to do with chess, could suddenly be encountered by a youth with an "incorrect" surname, and how his sports career could be imperceptibly hindered or even altogether ruined. My fears were justified: much has now been written about the latent anti-Semitism, especially in the upper echelons of Soviet Power. I am convinced that Garry Weinstein would not have got through to a world championship match with Anatoly Karpov either in 1984 or in 1987. They would not have allowed it." (45-46)
So Kasparov knew that his Jewish-Armenian descent, coupled with his connection to a far-flung region of the USSR (Azerbaijan) meant that he was the outsider. You may know the 1984 world championship match against Karpov went on for months, and was eventually called off by chess' organizing body, FIDE. As a result, Karpov retained his title. Kasparov is convinced that this was a conspiracy between FIDE and the Soviet government, to prevent him from becoming world champion. In Child of Change, Kasparov alleges that the President of FIDE "intervened with the connivance of the Soviet chess authorities to save their champion..." (2). Kasparov later elaborates that "My thesis in this book is that the arrival of Anatoly Karpov, who inherited the world title without having to struggle for it, coincided with a period in Soviet life that was marked by bureaucratic inertia and corruption. He associated himself with conservative forces in an alliance designed to maintain power for himself and for those who clung to his coat-tails. He ruled like a Czar of chess and resisted, by fair means or foul, any bid to usurp his throne. In doing this he managed to borrow the full apparatus of power of the Soviet state." (5).
Now Kasparov is prone to hyperbole, and you do have to take what he says in Child of Change with a huge grain of salt. Kasparov equates Karpov to Sauron, for example. But he certainly provides an explanation, at least from Kasparov's perspective, for why the Soviet chess establishment sought to prevent him from becoming champion.