Why was Winston Churchill so seemingly obsessed with the "soft underbelly" of the enemy? He backed both the failed Galippoli invasion in WW1 and the underwhelming Italian Campaign in WW2.

Elliot Roosevelt (FDR's son) wrote a book called "As He Saw It", which describes in detail the relations between Roosevelt and Churchill during the war.

Roosevelt thought that Churchill was "needlessly obsessed with power politics, too rigidly anti-Soviet, too colonialist in his attitude to what is now called the Third World, and too little interested in building the fundamentally new international order towards which American idealism had always tended".

Controlling the Mediterranean was vital for Britain's empire at the time, since it was the shortest route to connect the mainland with their colonies in Egypt and India. Indeed, they actually had a significant presence in Gibraltar, Malta and Cyprus, as well as in Greece (before it came under occupation).

Churchill had also stated numerous times to Roosevelt (and to other associates), that the Allies needed to cut off the advance of the Red Army into Europe. The initial plan was to push into North Africa and reach Egypt, then invade and liberate Greece, then head into the Balkans and try to make gains in Hungary, Austria-Germany and possibly south Poland, before the Russians reached those areas.

According to Elliot, FDR didn't agree to this for the reason stated above in the quote (and continued to say that he wouldn't sacrifice American lives for Britain's geopolitical interests), and instead they compromised to do the North African invasion, but invade Italy instead of Greece. That way they had a chance to knock out the strongest ally to Hitler in Europe, liberate French colonial territories and also have a foothold close to Germany. It would also force Germany to spread its forces in 3 fronts (when they would also invade Western Europe, as was promised to the Soviets), and it would provide a safer route to supply the Soviet Union. The main aim was to end the war as quickly as possible.

When the Italian campaign failed, they tried to do the same thing (have a foothold with quick access to Germany) in Belgium/Holland, then drive into North Germany and liberate Denmark, then make a land corridor, with naval aid from the Baltic, to Berlin. It would again help open a quick supply route to the Soviet Union. This failed as well, so having lost the element of surprise, and since the Soviets were already liberating Poland at this point, they decided the most sensible way was through France.

EDIT: This is part of a speech Churchill gave on the 8th of December 1944 arguing again for an intervention in Greece to save it from communism:

The House will be glad that I now come to Greece, which forms the mainspring of the Vote of Censure we have to meet to-day. I have taken great responsibilities for our foreign policy towards Greece and also in respect of what has taken place in Athens, and my right hon. Friend the Foreign Secretary and I have worked in the closest agreement. On or about 16th August, it became evident that the magnificent advance of the Russian Armies—[HON. MEMBERS: “Bolsheviks.”] Oh, no. That is a very mischievous remark. Some hon. Members are always trying to entrap me, when we have to have these difficult Debates on foreign affairs, into saying something which would seem to be disrespectful to the splendid patriot Armies which have cleansed the soil of Russia.

I say that we have taken great responsibilities, and when in August it became evident that the Russian advance along the Black Sea shore, and their probable impact upon Romania and Bulgaria was imminent, and this taken together with the advance of the British and American Armies up the Italian Peninsula, and also with the growing power of Marshal Tito and his partisans—whom we have always supported—would make the position of the Germans in Greece untenable. I therefore proposed to the President that we should try to gather forces to enter Greece as and when the German position was sufficiently weakened, and, above all, to save Athens from the anarchy and starvation which threatened it. I pointed out that, if there was a long hiatus after the German authorities went from the city before an organised Government could be set up, it was very likely that the E.A.M. and the Communist extremists would attempt to seize the city and crush all other forms of Greek expression but their own.

Churchill admits here that he wants to invade the Balkans, not to fight Nazis, but to prevent the Resistance movements of those countries, that are mostly led by communists, from taking over these countries and joining the Soviet Union.

The British Field Marshall Richard Carver (The Imperial War Museum Book on the War in Italy 1943-1945) mentions that the Americans and British had a big disagreement on how to approach the war in Europe. The British wanted a peripheral war, that would allow them to utilize their navy without having to fight on land, while the Americans had wanted from the start an invasion in France. They saw that as the path of least resistance, since the population would be friendly to them, and they would immediately be able to invade Germany. The British insisted that that would be suicidal. They compromised on the path North Africa to Italy, as intelligence suggested the Vichy soldiers would not fight the Allied forces, and it provided the Americans an opportunity to fight Nazis on a limited scale and with adequate naval support. However, Carver does mention that the motives were not purely strategic, and that international politics and post-war plans played a major role in the lines each side towed.

HP Wilmott (in his book June 1944) backs Carver's point of view, and further elaborates on Churchill's anti-Soviet/anti-communist stance. While Churchill couldn't express those views openly to the public (at least as far as the Soviet Union went), he tried to maneuver Britain and the US into a position that would ensure that the Soviet Union would be left in a weak position after the war, such that it would come to depend economically and thus politically on the Western Allies.

Churchill even ordered for Operation Unthinkable to be drawn up. These were two war plans that would strike at the Soviet positions after Germany's surrender, so that they would be pushed back, and Britain would be able to force their own terms on the table.

Even people that are biased in favour of Churchill's image as the savior of Europe, like David Reynolds (The Soft Underbelly), agree that Churchill's approach on the Mediterranean was based on imperial/colonial ambitions as well as anti-Soviet ideologies.

To answer your question, the decision to focus on the Mediterranean during WW2 had many motives behind it. There were clear military motives, such as knocking out Italy, opening an easy supply route to the Soviet Union, opening a second front, and then a third in France. There were also political motives, such as the Allies needing a victory to boost morale, the promise to Stalin to open a second front and helping out France, as a former Ally. However, there were international power play politics at hand as well, primarily stemming from Churchill and his fear that communism would spread.