Why did Churchill push so hard for the campaign in North Africa in WWII?

Mainly for political reasons, to keep the alliance together.

Eisenhower told Churchill that the attack in Europe would take two to three years to prepare, and that it was the only thing that could win the war against Germany. That meant Russia would be facing the entire German military, which Stalin was unwilling and perhaps unable to do. It meant the British people had to put up with more years of privation, casualties and bombings; it meant the U-boat campaign continued; and foreign allies would fall away, and German allies, mainly Italy and France, but also co-belligerents in central and northern Europe and the middle east, would stick with Hitler.

On the US side, people would start asking why we were bogged down in Europe doing nothing when it was Japan that bombed Pearl Harbor. If the US had lost the battle of Midway, for example, a lot of people would have said the US needs its entire military in the Pacific.

Plus people all over the world were saying let the dictators kill each other on the German Eastern front, who cares if continental Europe ends up fascist or communist? Getting US troops engaged in combat would solidify US support.

Eisenhower opposed the operation at first, but later conceded that the political calculations were probably correct.

German industry needed oil and North Africa was a gateway to the oil fields of the Middle East.

The Suez Canal was the gateway to the British Empire.

World War II: North Africa Campaign

The battle for North Africa was a struggle for control of the Suez Canal and access to oil from the Middle East and raw materials from Asia. Oil in particular had become a critical strategic commodity due to the increased mechanization of modern armies. Britain, which was the first major nation to field a completely mechanized army, was particularly dependent on the Middle Eastern oil. The Suez Canal also provided Britain with a valuable link to her overseas dominions–part of a lifeline that ran through the Mediterranean Sea. Thus, the North African campaign and the naval campaign for the Mediterranean were extensions of each other in a very real sense.

Also, after the fall of France, here was an opportunity for Britain to combat Italy and then Germany. As Nelson said: Close with the enemy!

Or, as Lance Corporal Jones liked to tell it: Sir! They don’t like it up ’em, sir!