You forgot an important part of the US government: The Electoral College. Even today, Americans don't technically vote for the president. Rather, we vote for electors who will vote for a candidate. Being that these electors have no functional purpose other than transmitting the popular vote, you never (as far as I know) see their names on a ballot.

Originally, the people had no direct role in choosing electors. Instead, electors were appointed by the state legislatures. This fit the general pattern of the U.S. Constitution sharply limiting popular participation in the federal government. Immediately following the Revolutionary War, men from the non-elite social classes flooded into all levels of government, much to the horror of traditional elites. These men agitated for reforms to the government to both make it more capable of dealing with national crises and keeping power in the hands of those that "best suited" to wield it. This movement eventually led to the Constitutional Convention and the creation of a federal government where only one branch, the House of Representatives, was directly elected by the public.

There was no plan for a party system, two-party or otherwise. Washington was actually opposed to the very idea of parties (an entirely sensible position). However, the formation of parties is somewhat inevitable - people will generally agree with like-minded people on most things, and any voting system works best if you have allies to vote your way on things. As such, there have been as many as four parties gaining electoral votes. The modern two-party system is not a natural law or an intrinsic part of American government, no matter how hard it tries to pass itself off as such.

So, in short, the House or Representatives was supposed to have a bigger role in Presidential elections, since the system was designed to have elections between several candidates chosen by a small number of people. Thus, the people's representatives would have a frequent and final say on who occupied that nation's highest office. However, directly involving the people was both undesirable and impractical.