Why do government's put up with fractional reserve lending?

  Because they realize they could not finance new economic growth by any other means than creating money from nothing. If the money supply was stagnant, so would the economy. The only people who would still have money are those who had already profited from the previous injection of new money. In order to create new opportunities, new money needed to be injected.

 Initially, government's were not so fond of such a concept. But they quickly realized that they fell behind nations that did operate a loose monetary supply and were also forced to adopt it.

 Much of this grows out of the gold standard. If gold is assumed to be the only form of monetary wealth. No one would work until they were paid in gold. Businesses and consumers could never complete a transaction until gold was traded between the two. But if the supply of gold remains stagnant, so would trade and economic growth. So banks devised a scheme where they distributed gold certificates instead of real gold. And thus the trade of these notes also known as currencies, helped encourage trade and economic growth. Because businesses and consumers can simply exchange these notes with one another for goods and services.

  There was much inflation from the introduction of national currencies, but at the same time, it encouraged an explosion of economic activities,  since these activities were no longer tied to the gold supply.