Did drawbacks of egalitarianism or mismanagement cause the rural famine in the Great Leap Forward?

Incompetency and deliberate malice

First of all, we should note that in years immediately after the end of civil war in 1949, China was still largely backward agricultural society. Apparently, only 3% of the population lived in urban settlements . Agriculture was based on small households with relatively primitive and obsolete farming techniques. Couple these with devastation caused by war and Japanese occupation, and it is not a surprise that food was not exactly plentiful even in these first years of People's Republic. However, situation was nowhere near as dire as it will become in few years latter due to communist meddling.

Stated goal of CCP was modernization of the country, which included industrialization and increased agricultural production. On the surface there is nothing wrong with that, but the manner it was done was artificial, plain stupid and even malicious. For example, traditional economic theories postulate that in country such as China industry should first follow needs of the agriculture, i.e. industry should primarily focus on producing goods that would increase agricultural output. Such increased output would then be invested towards infrastructure, education and further industrialization.

Note that Chinese communist were not satisfied with such "capitalist" theories. They wanted everything and now - rapid industrialization, modern agriculture, strong military and from beginning of Khrushchev Thaw they strived to reduce dependence on USSR due to ideological divergence and latter Sino-Soviet split. In theory, main difference between Maoism and more mainstream Marxism (Stalinism and Leninism) is in a problem of revolutionary class. Marxism states that working class (proletariat) is a main drive for change. Maoism instead relied on poor peasants, since China didn't have meaningful number of traditional industrial workers. However, in practice, Chinese communists wanted to transform and remold these enormous masses of peasantry into modern socialist (and latter communist) society, even if that meant millions of them would die. Attitude of Chinese communists towards its own population were best described by this quote of Mao Zedong

“I’m not afraid of nuclear war. There are 2.7 billion people in the world; it doesn’t matter if some are killed. China has a population of 600 million; even if half of them are killed, there are still 300 million people left.”

This is obviously about nuclear war, but it shows mindset pretty accurately.

With such mindset, CCP proceeded with destructive policies. It should be noted that collectivization in China begun somewhat before of Great Leap Forward, i.e. already from 1950 some land reforms were enacted. However from 1958 collectivization became massive and compulsory, similar to previous Soviet collectivization and similarly disruptive. Agricultural output dropped, but at the same time due to attempts of industrialization more food was requisitioned to feed growing army of industrial workers and other city dwellers. To add to the burden, villagers were often forcibly put to work on projects like digging canals for irrigation. Communist were prepared to sacrifice thousands in such projects as this Mao quote witnesses :

Wu Zhipu claims he can move 30 billion cubic metres; I think 30,000 people will die. Zeng Xisheng has said that he will move 20 billion cubic metres, and I think that 20,000 people will die. Weiqing only promises 600 million cubic metres, maybe nobody will die.

Overall, collectivization, forced work, employment of new and supposedly modern agricultural techniques (in fact done with little bio-technological knowledge) created Great Chinese Famine of 1958-1962. Countryside was devastated and millions died. Mao was forced to back down a bit, although he arranged comeback with Cultural Revolution. It should be noted that this first wave of industrialization was mostly a failure - due to the lack of even basic industrial knowledge failed projects like backyard furnaces were created. Overall, Mao Zedong and his accomplices cost China at least 30 years of development and likely 100 million deaths.