The last sentence here, I think, reveals the fallacy in this argument. It's the aisle-crossers who control the party as_an_institution_. They're like the tiller on a boat -- an inch this way or that, and you've tacked. Or gybed, as the case may be.
It's true that if you average up the (expressed) views of Democratic and Republican officeholders you end up with two different-sounding songs. But all the Bernie Sanderses and Dennis Kucinich-es and Ted Kennedys etc ad soporem are in effect lashed to a chariot whose reins are firmly in the hands of the Lantoses and Liebermans. So the ineffectual enlightenment of the former is worse than useless -- it's an actual snare and delusion, like the sweet nectar that draws the poor fly into the flytrap.
I like to think of the two parties as being a lot like McDonald's and Burger King. In practice, they're marketing the same thing, but they're going after slightly different demographics and have slightly different marketing and branding strategies, and slightly different Secret Sauces to mask the flavor of the same low-grade beef.