Let's clear a major misconception; your question is all about how it can be "right" to go to war, so now I'm morally obliged to counter it with a long, subjective and off-topic rant. Wars are not fought to immediately improve living conditions of a population. Moreover, they are not fought to annihilate the enemy, to wipe them off the planet, or to defend oneself from such fate. The primary reason always was, is, and will be, pure need of confrontation.
War usually does not come as a painful cold-blooded decision, like "too bad, we agree to loose some lives, but we really need the oil to be cheaper" or "... but we really need to live more comfortably" or "... but we really need to free those people from jails and concentration camps". Instead, the war erupts and escalates quite suddenly, without much consideration of future sacrifices and effects.
Humans and some animals (this is male thing in most cases) very much like to risk our own comfort and our own life in a kind of pre-staged situations. Historically, primary example of this is an armed conflict. This need is so strong and prevalent, that even in modern times, when we are educated enough to surely predict that innocent bystanders (even women and children!) would get hurt, from time to time we cannot stop ourselves, which by the way pretty much makes us a bunch of quite murderous beasts, doesn't it? Well, too bad, but to the point.
- Enter 1939. Germany wanted to confront Poland. They surely could leave them alone and concentrate on defending their land and living there happily. They choose confrontation!
- Britain and France wanted to confront Germany. They surely could leave them alone and concentrate on defending their land and living... etc.
- ... etc. (aren't all the following scenarios similar?)
- Japan wanted to confront United States. They surely could ... etc.
- Here we are in December 1941, end of the rant, answering your question, please forgive all the rambling:
There are three teams in the world now, Axis, Allied and neutral. One of the Axis have already confronted US, so nevermind US was leaning towards Allies previously, it doesn't really matter. Now US public wants to confront Japan and see them dominated, US commits themselves to Allies. But the leadership, both British and US, is not stupid. They know they can dominate the enemy more effectively (to "win the war" more easily), by playing as a team and by the principle of concentration of power. Why to go one-on-one each in their own theater? They decide to overwhelm the enemy team by concentrating mainly on one region and then on another, etc, until enemy surrenders. Note, the ultimate aim is to dominate the enemy, not to gain land&resources in some theater. On the contrary, they would gain land&resources just as a convenient way to show their enemy "here, see, I've just confronted you and I've dominated you".
Churchill promptly visits Roosevelt for "ARCADIA" conference in December 1941 and they start discussions. Italy is the weakest member of Axis team, can all Allies concentrate on them first? No way. Well then, Japan? Not so good, both Britain and USSR are weak on the Pacific (The World War II Conferences, pp. 17-40). Well then, mainland Europe? Obvious bold yes expected from USSR, but US and Britain are too weak yet (for 1942 and 1943 plans, see Operation Sledgehammer and Operation Roundup respectively, US Army in WW II, pp. 8-10). Roosevelt knows US public is more against Japan but would ok'd anti-German war. Maybe Roosevelt thinks that Germany would dominate USSR, maybe he doesn't - it's irrelevant to my answer. What is relevant, the USSR is fighting Nazi Germany as of now.
Britain eventually proposes to go against Germany outside Europe, to start in North Africa where Germans are weak. Operation Torch ensues, including US landing in Africa. Soviet Union is somewhat disappointed, but they realize that, in terms of all-Allied power concentration, it's the most favorable scenario (North Africa does divert some German resources from USSR theater).
In terms of global "Axis vs Allied" thinking, it doesn't matter for Roosevelt if there is immediate economic US interest in Normandy, or in North Africa, or in Antarctic for that matter. Axis try to dominate Allies, Allies try to dominate Axis, whatever it takes.
The Allied invasion on Sicily follows as a logical consequence, so does the Allied invasion in Normandy.
EDIT: In support of this, count of US Army divisions in active combat. At the very beginning of US invasion on Tunisia 6 divisions, while 4 on Pacific. At the very beginning of US invasion on Philippines (Leyte, October 1944) 17 divisions on Pacific, while more than 30 in Europe. chart