This question needs some context in order to be answered and understood. While elements of the KMT split between centrists, rightists and leftists never got along, they were able to function well enough to initiate the 1926 Northern Expedition, which is very early in KMT history.

Part I

First, its very true that many members of the CCP did not wish to cooperate with the KMT, and vice versa. In fact, Sun Yat-sen was quite leery of including the CCP among the KMT in the early 1920s. They did so because they were forced to by the USSR, who was funding the KMT at this time. Any and all of the success of the KMT from 1919 to ~1941 relied on the bulwark of materiel and financial aid the party was receiving from the USSR, who perceived the KMT as a potential ally in the fight against Western imperialism. This aid only increased as Japan became more powerful, and the USSR was hoping to turn the China into a buffer state to prevent future Japanese expansion in East Asia. So without including the CCP by demand of the USSR, none of that rapid success the KMT experienced under Chiang would've happened. Combine this with an increasing existential threat coming from Japan, and that should answer any doubts as to why the United Front stayed together for so long despite a longstanding civil war.

It became increasingly clear to Chiang and other party members that most elements of the CCP were becoming radicalized throughout the 1920s. Even Wang Jingwei, the liaison between the leftist and rightist elements (and future collaborator with the Japanese in China), was distancing himself from the radicalized leftists of the CCP. There was even some fracture among the CCP itself. Founders of the CCP such as Chen Duxiu and Li Dazhao were committed to different goals than the younger and more radical elements that arise post-Shanghai Massacre. I'll discuss later.

Now, I feel like this question assumes Mao was important for the entire history of the CCP. This is not exactly true. Mao's coming to power isn't solidified until the later 1930s, a decade later, during the Yan'an years. He was certainly out there, working as a sort of lower ranking propaganda officer in the KMT military, but the power of the party in the early 1920's was in Chen Duxiu and Li Dazhao. Mao was recognized as important to the party; he came from a wealthy peasant family in Hunan and he was ambitious and well-spoken as well as organized.

That brings us to April, 1927 and the infamous Shanghai Massacre. This is generally used as the beginning date of the Civil War that ends in 1949. There was a lot of confusion in the midst of the massacre as it took many CCP leaders by surprise. And this is where Mao initially rises to prominence. In the aftermath, the CCP established a base in Wuhan and began sending feelers out to surrounding provinces, trying to establish support for the CCP and revolution against the KMT. Mao, being an accomplished young man from a wealthy family in Hunan, was sent there and it was here he wrote one of his first major pieces on communism: Report on an Investigation of the Peasant Movement in Hunan.

Part II

The writing of the report is, in retrospect, quite momentous. At the time it was disregarded by many older leaders in the CCP. Throughout the summer of 1927, Mao spent time in rural Hunan, reporting on what the peasants complained of. How much of the report is true and how much embellished can be debated, but ultimately the conditions were, not surprisingly, awful. Caught between constant conflict and being economically deserted by the KMT, the rural peasantry of Hunan (and, one could assume, China) had plentiful complaints. In September, 1927, Mao organized and led the Autumn Harvest Uprising. These two things, the report and the uprising, are Mao's responses to the betrayal.

What's so momentous about the report and the uprising is how it shifted the ideology and leadership of the CCP. To address ideology, Mao stated that the CCP should stop focusing on urban centers and shift their backing of the rural peasantry. This is quite unorthodox for Marxism in general; Karl Marx himself famously referred to the peasants (of France) as a "sack of potatoes." This is where Maoism really forms as an independent ideology. Chen Duxiu and Li Dazhao had both espoused the Marxian and Leninist practice of placing importance to the urban proletariat, while abandoning the rural peasant, who symbolized the old and feudal, submissive to their lords. Mao did not believe the CCP could succeed without the support of the rural peasantry. The peasants were just too important to Chinese history and society.

The Autumn Harvest Uprising signified the shift in leadership of the CCP from the older communists to younger, pro-Mao leaders. In 1931, Mao and Zhu De formed the Jiangxi Soviet under their rule, practicing Maoist ideology. Mao was armed with a clear and organized ideology as well as supporters. The older communists had never been able to create any sort of definitive text outlying what the actual ultimate goals of communism in China were. Mao did, however, and his increasing hold over the CCP would lead to it becoming a Marxist-Leninist style vanguard party during the Jiangxi and Yan'an period. As early as 1937, but certainly by 1942, Mao was the indisputable leader of communism in China, and a new key ally of the USSR against the KMT, who proved to be less loyal to the USSR than Stalin had hoped.

To recap, for the 1927 betrayal, Mao's response was the total abandonment of KMT cooperation, leading an uprising in Nanchang, establishing a Soviet-style republic briefly, and consolidating his hold over the CCP as its indisputable leader. This is the first phase of the Civil War, from 1927-1936. In 1936, Chiang is captured and forced by internal and external pressure to recreate the Second United Front, which lasted from 1936-1945. In 1945 the Civil War resumes.