Would the United States have been better off in the long run if Monroe had joined hands with Britain to keep the other European powers out of the America's?

no. it wouldnt have

the expansion of the rule of empires over other territories took place as a result of several reasons including economic, ethnocentric, political, and religious, and exploratory purposes.

in terms of economic purposes: imperial governments aimed to maximize profits and with that purpose they neeeded cheap labor, access to or control of markets to sell or buy products. they also desired to find natural resources (precious metals and land). in the awake of   industrial revolution, dependent colonies provided the necessary raw materials that european factories and market needed to manufacture products.  

ethnocentric: many territories were conquered by imperial territories in an attempt to spread their cultures, values and belief as they considered them to be superior. in the late 19th century, for example, european powers held   the racist belief that inferior races should be conquered in order to “civilize” them.

political: another reason that drove conquer and expansion was patriotism and growing imperial power. it refers to national prestige, pride, and security. empires sought strategic territory to ensure access for their navies and armies around the world.

religious: the aim of converting new member of other religions was another factor that prompted expansion.   christian missionaries from europe established churches in conquered territories during the nineteenth century. thus, they spread western cultural values. furthermore, the imperial nation’s language was spread as well, through educational and religious interactions.

exploratory: imperial nations or their citizens aimed to explore territory that was unknown to them,   for different purposes including medical or scientific research, however, at the very   beginning, they did it to embark on adventure. furthermore, imperial explorers sought to discover, map, and claim territory before their imperial competition did, for national and personal glory as well as to realize the imperialist goal of expansion.

d. migration