"As for the socialist side, they say that the whole reason for stagnation was that capitalistic reforms (such as Kosygin Reforms) had taken place and that the economy slowly turned away from the centrally planned economy mainly driven by Five Year Plans."

If anyone is saying this then they are vastly overstating the importance of the Kosygin reforms, which did not tamper much with Soviet central planning, and were relatively quickly abandoned.

If we're talking about serious reforms to the Five Year Plan central planning model, that only really happened in the later stages of Gorbachev's perestroika, which I discuss here. That's really the point where the Soviet economy began to unravel. Prior to that, the era of stagnation mostly meant ever-slower economic growth, but that's not the same as zero economic growth or decline (although depending on how the statistics were calculated the USSR may have had a short recession around 1980). Interestingly, it was Gorbachev himself who popularized the phrase "Era of Stagnation" to describe his predecessors' economic management compared to his plans.

As for the "capitalist" side: a lot of writers on this end of things will say that the Soviet economy was doomed to fail, often through its own inherent internal contradictions (which is a bit of a Marxist way of looking at things, ironically, but anyway). This is correct in the sense that the Soviet economy never really functioned in the way it was supposed to on paper. As I discuss here, a massive part of what kept the command economy working was through state enterprises having informal "pushers" who often bartered with each other for needed inputs, and who influenced the planners to adjust their targets.

In addition to this, the Soviet economy's ever-slower growth rates impacted and were in turn impacted by other major problems in the Soviet economy. Briefly listed: an agriculture sector that under Khrushchev had seen major improvements in terms of workers' pay and benefits, but no real improvement in productivity, with the USSR eventually needing to import grain, including from the US. A consumer goods sector that because of planning wasn't really able to respond to consumer demands, and was completely insulated from quality concerns by consumers. This was contrasted by an extremely large defense industry that was heavily prioritized and focused on the needs of its (one) major consumer. Central planning also had linear concepts of investment, ie state enterprises were expected to invest x% of their resources into new assets every year, but it was always much easier to just to more of the same, ie get another existing type of factory machine rather than invest in the R&D for a completely new type of machine or production process. Finally there was the massive wastefulness involved: state enterprises were concerned with hitting their periodic production targets above all else (it's how managers were rewarded or penalized), but effectively they were only interested in just hitting that target. You got rewarded just the same for making 1,001 Ladas in a month as you did for making 10,000, if 1,000 was your production target, and you got penalized/not rewarded the same for making 999 or 500. Efforts to tinker with incentives (as the Kosygin reforms tried to do) never really managed to fix this, nor did any of this really address whether you'd actually get the inputs needed to make 1,000 Ladas (maybe you'd get too much, or too little, and either way your pusher would be working the informal market as a result), nor did the central planners care about things like massively subsidized energy costs, or the fact that enterprise managers could use the basically-open labor market to just massively hire people to produce to meet immediate deadlines (which also never really addressed issues of productivity).

So I guess I'd say neither "side" is 100% correct, but both have some kernels of truth. The "capitalist" side is right that the postwar Soviet economy saw increasingly lower rates of growth, never really functioned as it was meant to, wasn't able to close the gap with advanced Western economies, and had massive areas of inefficiency. But it was stable - it wasn't really in imminent danger of collapse. The collapse ironically happened by Gorbachev's attempts to reform the centrally planned system, which mostly undermined it (and to be clear here, Gorbachev wasn't actually trying to create a market economy or capitalist economy per se, but he undermined the existing centrally planned economy enough that effectively everything went into freefall).