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The "social problem" was the Communist influence in China, and ultimately the right-left split in the GMD. The reason that Shanghai was important was because the Communists were heavily concentrated in the city. So there were two GMD "attacks" on the city; the first, against the warlords that held the city, and the second, against the Communists and their allies, the labor unions. The Shanghai incident essentially wiped out the urban Communists, and only Mao tse-tung's peasant orientation allowed a revival.
This forced the left wing of the GMD to abandon their support of the Communists, and (temporarily) re-united the GMD. An attack on Beijing first would have represented a continued attack on the warlords, that would have left the issues of Communist involvement in the northern expedition and the GMD's left-right split unresolved.
As for Chiang's allies, they were the Chinese "Mob" and their leader was Chiang's personal friend (the two had been "drinking buddies in their late twenties and early thirties). They wanted to get control of the labor unions, and Chiang wanted to eliminate Communist control over them.
To paraphrase Chiang himself (in a slightly different context), "The warlords are a disease of the skin, the Communists are a disease of the heart," the latter being the "social issue" Chiang cared most about.