All one-party states tend to be corrupt, since there are no checks and balances or any reason to have any oversight. So the idea that Kh and his family were corrupt were just politically expedient and easily palatable horseshit excuses to displace him, especially considering that Brezhnev family, with especially his monster of a daughter, were hundred-fold more corrupt than anything Kh did with his brood. The problems were:
1. By decentralizing the industry and giving more "power" to the locales, he in effect sent hordes of Moscow officials in charge of planning and industry into exile by having them leave the comfort of Moscow and all that entailed and had them go to into the provinces to supervise from there. In the 1950s, the drop from creature comforts of a city like Moscow to a town in the middle of the Urals was steep indeed, and we're now talking thousands of disgruntled officials and their wives. And all this done without consulting or even discussing. In a one-party tyranny, the bureaucrats cannot protest of course, but they can screw-up accidentally on-purpose. Anyone who has ever worked in a corporate environment has seen the power of someone being able to gum up the works through "legal" means if they are disgruntled.
2. Decentralization caused lines of communication to be confused, causing chaos and eroding the power of the ministries that were initially rather supportive partners of Kh in terms of his "coalition."
3. No blood-letting of the anti-Kh members. In a one-party tyranny, fear counts for a lot. During the first anti-Kh putsch, when Malenkov and Molotov make their move with half of the Central Committee out of town, Kh demanded a full Central Commitee vote, and M&M agreed, thinking they had the majority. They didn't. Kh had the Soviet Air Force fly in the pro-Kh Central Committee members and Kh won the day and expelled the troublemakers, but did not have them arrested or executed. Merely demoted. It meant that when Brezhnev and his pals were gathering for a coup, they knew their price of failure was not a bullet in the back of the head, but being made factory manager in Magnitogorsk. The instruments of fear were removed by Kh and he could no longer apply them or have people fear them.
4. Reduction of the Soviet Army. All armies have bloat, but none were as bloated as the Soviet Army in the 1960s. The issue wasn't soldiers, who were a conscript force just passing through and who soldiered for a few years and then were made into "productive" members of society, churning out sprockets at Lenin Works Factory Eleventy when they'd get demobilizing. The issue was the officer class, for all those infantry and artillery regiments. From the point of view of the state, a horde of people who had to be fed and clothed and whose families likewise had to be billeted and taken care of for ever and ever, and who were not "productive" and who upon retiring early would then just sit on a pension and not do anything. The issue was so serious that there were rules restricting which towns were allowed for officers to retire in, because there was a fear these un-productive members of society would take up too much space in the "good" towns and displace productive members of it, leading to a drain on the resources there. But hey, thank goodness for innovations, because here come rockets, right? And rockets are cheaper than a horde of regimental officers and regiments of soldiers as well! But, uh, reducing the army to save money, and the victorious Red Army which Won the War is not good politics. Also, the first time around, it was the Air Force who bailed out Kh during the Molotov putsch. He was biting the hand that fed him, even if he was doing it for sound economic reasons.
5. Virgin Lands fiasco. Kh had a plan. Well, he had a lot of plans. But he was the "agriculture" guy so naturally he had an agricultural plan. Virgin Lands out in the East, which would be colonized and farmed and harvested and be awesome. The problem of course is that most of the Soviet Union is drought-heavy as compared to the rest of Eastern Europe, never mind good soil of Western Europe. 1 in 10 years in the Soviet Union, on average, were drought years. In the Virgin Lands, it was one in seven, a bit better. But in comparison, IIRC, in Poland, it is 1 in 100. So the chance of failure was much higher and it hit. 1963 was a harvest failure year. A public slap in the face from Mother Nature to Kh and his agricultural policies. Also, the Colonization of the Virgin Lands was a massive undertaking, and the amount of resources devoted to it - bearing no yield - was costly in money and prestige.
6. 1962 Party reforms. Kh hit upon the novel idea of getting "fresh blood" into the Party by having a third of the party committees replaced every one to two years. This of course pisses off the regional secretaries who now lose patronage. If one third of the jobs are up for grabs every couple of years via even horseshit elections, then you cannot guarantee the jobs for your cronies to be set for life, which goes at the very heart of the bureaucratic regime.
7. 1962 Party reforms, part two. Kh wanted to split committees and specialize them. He particularly like the notion of having agricultural committees and industrial committees be separate. And this of course caused bloody chaos for two reasons. First of all, which of the committees will be in charge of police? Who will run education? What about health? All these things have budgets. So it's not just the question of responsibility, it's a question of power, money and patronage. Also, this brought the hermit and secular-Soviet monk-saint Suslov out of his one-bedroom tiny apartment full of books of Marx and Lenin and blink at the sunlight and wag his bony finger at Kh and rally the faithful. By splitting the committee, is not Comrade Kh now advocating a split in the very Party itself? Is he not advocating the division of Industrial Party and Peasant Party? He is spitting on Lenin! Suslov was widely admired and respected as a true-believer. For Suslov to speak out against a Soviet leader, was like for a Cardinal to call for the downfall of a corrupt Pope. And it gave gravitas to the band gathering to remove Kh.
Kh was Kh, so his high-handed manner and abrasive personality were not going to change. Neither were his family obligations. And agriculture will defeat anyone in the Soviet Union if they hang around long enough, due to drought and shitty Soviet economy and piss-poor planning of a one-party tyranny. But alienating the Army and the Party could have been avoided, as could have shifting all those bureaucrats around, maybe. But this would have drained the economy.