See, that happens in the UK when the parties choose their leaders - just without the messy touring and without the necessity for people to actually come out and vote. The Tory Party (and the LibDems, I think) actually get their members to submit postal ballots to pick the party leaders, which seems to be, in essence, the same as the Primaries but cheaper, quicker and less effort.
Not only that it costs less effort, but I have a more fundamental gripe: a party is an association, and if you want to have a vote who is to become its leader, you should join as a member! It's not as if it's very expensive to join - at least not over here, the party I'm member of asked a minimum contribution of EUR 15 per year.
Here in the Netherlands, traditionally party leaders are elected at the party congress, by representatives of the districts. Recently, two major parties switched to a direct vote by all members, which of course garnered quite some media attention (especially the "American" campaign of
Ms. Verdonk).
In the UK, the people do not pick the head of state. Different critter, totally.
But in the UK (and in Holland) the Queen is head of state, and has only a ceremonial function. The party leader in the UK of the winning party (nearly) always becomes PM and thus political head of the government.
But ultimately, the party leader is there to garner as much votes as possible for his party and thus get its party into government. I don't see much difference there between the US system which directly chooses its head of government, or the US or Dutch system where a government is formed on basis of parliamentary elections.
There is a second problem I have, though, with the primaries. In most states, you have to register as a "Democrat" or "Republican" voter to be able to vote in the respective primaries. Isn't this counter to the principle of secrecy of the ballot? Of course, you don't have to vote according to this registration, but why should the government register someone's preference for either party? In my view, that's up to the parties themselves to have a registration of voters who want to vote in primaries.