If the top two in the primaries are conservative republicans then what? In some states that may well be the case. Nationwide that would likely depend on the turnout in the primary as opposed to the turnout in the general election.
The two candidates that make it to the general election would probably have a similar ideology, but that is a good thing. They'll be representing the real center of the population and it will be a competitive race in which the two will have to compete for ALL voters.
Right now, most districts are a lock for one party. You may have a Democrat and a Republican facing off, but that is irrelevant because it is guaranteed that the dominate party will win. It is basically just a show election in which the incumbent only has to answer to his base.
Let's imagine the worst case scenario in which you have a swing district in which one side has low voter turn out and you get two candidates who don't quite represent the center of the population. It is still going to be a competitive election and both candidates would still be more moderate than what they would have been if they had been picked by their base alone. The two candidates still need to win over as many voters as possible and the eventual winner still has to worry about being reelected next time in an environment that is much more competitive. In addition, if the winner completely marginalizes the side that didn't show up in the primary, that will likely energize that side to show up next time.
Either way you slice it, you still have to worry about turnout in the current system and in the system I propose. You could argue that lower voter turnout in primaries now indicates that there would be lower voter turnout in the approval primaries. That doesn't make much sense though because there are completely different circumstances surrounding the two scenarios.
Right now, the general election is the much more important election and the primaries generally exclude independents. Regardless of how the primary turns out, you are still going to get a Republican and Democrat. Even under plurality top two, you will still probably get a Republican or Democrat. Under a non-partisan approval primary, this is reversed. It is the primary that is the most important election and the general election is just a fine-tuning. It would make more sense for there to be higher turnout in the primary than the general.
The exception to this might be during the first couple elections if voters don't even realize that the primary procedure has changed.
IMO to get out of the first past the post system, and have more than 2 parties we'd need to go to a Parliamentary system, with nationwide proportional representation. This would likely make 3rd parties viable, at least as far as the need for coalition government. I highly doubt that would happen.
That would require a constitutional amendment. Non-partisan approval primaries could be done state by state. The Presidential elections would still be a problem, but I think the best solution for that is to try to get a bill passed in each state that allows approval voting to be used in the general election and allow the winner to determine who the State's EC delegates would be obligated to vote for.
