"[I may be ignorant] but it sounds like there might be something to be gained by getting rid of a government that works this way."
The U.S is a fairly unique contry, politically. There are many other political systems in use by socially and economically well-off countries, but none have many of the most corrupt-able features that the U.S has. It's almost certain that any form of political system would lead to a government more like what you could see in these other countries.
Two point summary of the differences: kinder socially, more economic interference. On the one hand, there's more concern with the welfare of people, on the other, there is a reason the richest people and corporations exist in the U.S.
"What would we lose that we need if we got rid of them?"
Americans are proud of being an "innovative" place. Much of that is finding new ways of exploiting others, but much is also the genuine drive of Americans to accomplish things, whether it's trying something new (inventing things, new business ideas, combining strange things) or just being flat out competitive without care for the consequences (WallMart, Microsoft).
By being better socially, people would be aware of how things they do affect others, and would hesitate or have second thoughts. This might save much misery, but would miss opportunities.
"And if it is a good idea to get rid of them -- I don't know if it is -- how do we do that? I mean, just replacing them with a differant party doesn't seem to help when you are looking at the bigger picture."
Ideally, you need to make government - the actual administration, which does the actual running of things - more responsive to the will of the people.
First, you need to allow people to choose without fear that their choice will be wasted. Some U.S counties already use "instant runoff" elections, where voters mark their preferences for candidates, rather than only one. First choices are counted. If their first choice is eliminated by having too few votes, that one is taken off the ballots and the choices are moved up one and counted again. This is done until someone gets a clear majority (> 50%). This lets someone vote without fear for someone who has a low chance of winning but who they really approve of.
Second, the people who run the upper levels of government should be responsible to the electorate directly. In most countries, they are selected from the country's parliament. If they displease the people, they risk their own jobs. In the U.S, they are only responsible to the President, and have no need to care what people think. Congressional approval was meant to counter this, but is almost entirely ineffective. It might be a constitutional conflict to choose members of Congress for these posts, but maybe the Electoral College may be more appropriate, as they elect the President anyway, they may as well be called upon to serve posts in government as well. College members would then have to campaign on their own behalf, in addition to the presidential candidate they represent.
I have mixed feelings about elected judges, but I tend to come down on the side of not electing them. Rules of qualification might be a better way.
These changes ought to be reflected in State governments too.
I think this would be a start. As for how to implement this, that's a doozy. Especially in a country where two Presidential candates were arrested last election for trying to petition for inclusion in the televised debate as required by law.