Um, the issue isn't who won the election. We know that under the rules as currently laid out by the Consitution, Kennedy won the 1960 election. Nobody disputes that. The debate is over what metric should matter in a healthy, functioning democracy. Those (like me) who support abolishing the electoral college believe that when it comes to choosing the one elected official who gets to claim to have a mandate from the people of the nation at large, the actual will of the voters should take precedence over a deeply flawed measure of how "widespread" each candidate's vote is.
I was agreeing with you until that last sentence. I still do not "treat" or perceive the President of the United States as "the representative of the people at large." That is not the constitutional or accurate way to look at what the President does. He's the Chief Executive. The President is to the national government what the Governor is to a state government. In almost every state, whoever gets the most votes -- first past the post -- wins the office of the Chief Executive. The same ought to go for the presidency -- not because the presidency is a representative of the people, but because he is the Chief Executive.