Well, the problem with cheap probes is that NASA doesn't want cheap. NASA wants a mission to be expensive, as high as they can get "without going over" to the point where Congress will cancel it.
NASA's purpose is to hire and pay themselves and their contractors. Actually exploring space is a distant second priority. This has been true pretty much since the first Shuttle started development.
Somebody besides NASA could certainly do cheap space probes, but it won't be American.