The primary argument for "Fix the money, fix the world" is that government issued debt-as-money, i.e. fiat, is a source of corruption. Corruption of financial, political, and moral incentives at the highest levels of society. Fiat money offers a mechanism for a few elite to teleport value from all current holders of the currency to their own pockets. With such power, they can pay for any agenda they wish to pursue, good, bad, or deadly. You can truly "rule the world" with fiat monetary policy.
Bitcoin brings to the table an alternative to fiat that is not susceptible to the theivery that, at its core, fiat is. When Bitcoin is adopted by a population, fiat producers can no longer pick the pockets of the masses who no longer value their inflating currency. They lose that golden goose that could previously pay for all their fancies. The suffocation of a class of people that use money to rule over mankind is a path to saving the world from their deadly influence.
I think that is the main problem Bitcoin solves. The problems of society remain, but a major source of them is removed. Removal of a deadly cancer doesn't make you immune to all other disease. The procedure lets you survive and fight the battles that follow.