Since the government does not have a magical tree of plenty and can only redistribute from the left pocket to the right by taxing, borrowing, or printing money, how does this make any economic sense or make any country better off?

Government and Keynesian economists will tell you it's to protect us from the coming dangers of hoarding; specifically, that banks will stop lending and just let funds sit. Keynes brought hoarding to the forefront of economics in his The General Theory of Employment, Interest, and Money; a concept that the classical economist considered to be irrelevant.

In a circular flow economy, the value of output must be equal to income. Income represents an ability to purchase goods and services and can be divided into three categories: it can be consumed, saved, or hoarded. Consumption is using income to obtain goods and services for current personal satisfaction. Savings is (correctly) defined as a transfer of purchasing power from one group to another.

The saver is giving up his current access to goods and services to be able to consume more of them in the future.