The founding fathers did NOT want a two party system and in fact warned against them. We failed to heed their warning and now see their fears come to fruition.
John Adams:
There is nothing which I dread so much as a division of the republic into two great parties, each arranged under its leader, and concerting measures in opposition to each other. This, in my humble apprehension, is to be dreaded as the greatest political evil under our Constitution.
George Washington:
The alternate domination of one faction over another, sharpened by the spirit of revenge, natural to party dissension, which in different ages and countries has perpetrated the most horrid enormities, is itself a frightful despotism. But this leads at length to a more formal and permanent despotism. The disorders and miseries, which result, gradually incline the minds of men to seek security and repose in the absolute power of an individual; and sooner or later the chief of some prevailing faction, more able or more fortunate than his competitors, turns this disposition to the purposes of his own elevation, on the ruins of Public Liberty
Without looking forward to an extremity of this kind, (which nevertheless ought not to be entirely out of sight,) the common and continual mischiefs of the spirit of party are sufficient to make it the interest and duty of a wise people to discourage and restrain it.
It serves always to distract the Public Councils, and enfeeble the Public Administration. It agitates the Community with ill-founded jealousies and false alarms; kindles the animosity of one part against another, foments occasionally riot and insurrection. It opens the door to foreign influence and corruption, which find a facilitated access to the government itself through the channels of party passions. Thus the policy and the will of one country are subjected to the policy and will of another.
Reading the above quotes it is easy to see that everything they said would happen under the party system, has in fact not only happened, but is actually getting worse. The jealousies, false alarms, animosity, corruption, and foreign influence Washington warned about is indeed happening as we speak. One can see all of this manifested in nearly any news article. We the people should see that it is indeed the great "political evil" Adams said it was.