The problem arises when 95% of the population is fooled into voting for a single party with two wings, both of which are working against them.
Frankly, I doubt you understand politics. Despite your claims the parties are different entities although with very similar goals (power and control). In some areas, the policies of the parties are indistinguishable because they are appealing to same people for funding and trying to get same people to vote for them. Both parties need a majority of votes to win so they are by necessity fighting over the same people in the American center.
Frankly, in the current American system, large differences are not sustainable because if the difference loses votes, it will be abandoned and if it gains votes it will be copied or mirrored by the other party. The American system, whether by design or by accident, generates nearly identical parties.
It's not that the parties are the same organization, because they clearly are not, it's that the American political system is so poorly designed that serving the people brings few benefits when compared to playing internal politics for advantages and begging money from sponsors to fund election campagins.