Rick Hertzberg quotes Alexander Hamilton arguing against a supermajority requirement: 

What he was attacking was the premise that would one day underlie the McConnell-era filibuster—the notion that a legislature should routinely require supermajority approval for any action to be taken. For one thing,
To give a minority a negative upon the majority (which is always the case where more than a majority is requisite to a decision), is, in its tendency, to subject the sense of the greater number to that of the lesser.
That’s bad enough in itself, but it becomes positively dangerous in times of serious trouble:
In those emergencies of a nation, in which the goodness or badness, the weakness or strength of its government, is of the greatest importance, there is commonly a necessity for action. The public business must, in some way or other, go forward. If a pertinacious minority can control the opinion of a majority, respecting the best mode of conducting it, the majority, in order that something may be done, must conform to the views of the minority; and thus the sense of the smaller number will overrule that of the greater, and give a tone to the national proceedings. Hence, tedious delays; continual negotiation and intrigue; contemptible compromises of the public good. And yet, in such a system, it is even happy when such compromises can take place: for upon some occasions things will not admit of accommodation; and then the measures of government must be injuriously suspended, or fatally defeated. It is often, by the impracticability of obtaining the concurrence of the necessary number of votes, kept in a state of inaction. Its situation must always savor of weakness, sometimes border upon anarchy.

It's clear that the Constitution specified a few key areas where a supermajority would be required -- treaties, overriding vetoes, and Constitutional amendments -- with the obvious understanding that other votes would be conducted by majority rules. The filibuster simply evolved due to the imprecision of Senate rules.

It's interesting, given the right's newfound interest in hewing to the spirit of the Constitution. There are two key ways in which parts of the government have usurped power in clear violation of the intent of the Constitution. One is the filibuster. Another is the enormous war-making power of the president, who the Constitution clearly intended to constrain from initiating conflict without congressional approval. Somehow conservatives show no interest in restoring constitutionalism in either area.