I basically agree with Pascal-Emmanuel Gobry about the failure of the "independent" central bank concept, but I think the question of why the politics of the price level have shifted requires a different explanation than this one:
Politicians are crankish about monetary policy for a simple reason: because they know that it's out of their hands anyway. It's like national politicians in Europe who demagogue about policies that are EU competency. They know they're not going to do anything about it, so they don't have to think about it, they don't have to know about it, and they can say whatever they like. It's interesting to note that, historically, the populist political position was generally pro-inflation--remember William Jennings Bryan and the Cross of Gold? Inflation squeezes creditors--bankers and rich people--and helps debtors--less-rich people. Instead and in defiance of political logic, populists in Europe and in the US are anti-inflation. This is because they don't have to think about monetary policy because they know they can't do anything about it. So they bleat about inflation because when regular people hear "inflation" they hear "higher prices = bad".
I think the real issue here is the changing structure of the economy, not the changing structure of central banking. William Jennings Bryan's constituency was farmers, which is to say sellers of primary commodities. If that's your occupation, then it's obvious that higher nominal prices means higher nominal incomes. Even in Bryan's day, urban workers were not in love with this message and today almost everybody is an urban worker and almost nobody is a farmer.
Now a fancy-pants economics blogger can tell you that the most important price in the economy is the price of labor and the price of labor is equal to workers' incomes, so a general increase in the nominal price level is necessarily a general increase in nominal incomes. But nobody seems to believe that. Instead people are convinced that gasoline and milk are the main prices in the economy, and that a general increase in the nominal price level is necessarily a general decline in real incomes. Worse, oodles of media coverage have gotten people confused between the monetary concept of a general price level and the lifestyle concept of a "cost of living." The cost of living is massively impacted by real scarcities. A bad harvest in Australia, a war in Libya, a draconian zoning code in Silicon Valley, or a fad for strained yogurt increase the price of wheat, oil, houses, and yogurt respectively but none of those things are inflationary monetary policy and none of them can be reversed by tight money. The issue is that there's genuinely less "stuff" to go around. People—rightly!—dislike it when the cost of living goes up, so insofar as they think that loose money is a cause of high costs of living they will naturally rebel. Those of us who write about these issues need to try to explain them more clearly, since I think the language that's conventionally adopted is genuinely confusing.