In the rest of the world, justice comes before anything else. No matter how evidence is obtained, it is still evidence, and will be used in court.
In the US, if an unskilled policeman makes a small mistake, all evidence will be thrown down the sink, and the criminals who would be convicted on that evidence in every other country of the world, will walk free. I don't understand this protection of people where there is evidence that they are criminals.
The rest of the world also has due process of law. In America, we do tend to take parts of it more seriously than some other countries; so much so that we have constitutional protections form the government (e.g. forth amendment to the constitution banning illegal search and seizure). This is not unique to the US, but the way that it is implemented, it is a little bit unusual.
It sounds like what you may be talking about is known colloquially as 'fruit of the poisonous tree'. The linked wiki article has a really nice example:
For example, if a police officer conducted an unconstitutional (Fourth Amendment) search of a home and obtained a key to a train station locker, and evidence of a crime came from the locker, that evidence would most likely be excluded under the fruit of the poisonous tree doctrine. The discovery of a witness is not evidence in itself because the witness is attenuated by separate interviews, in-court testimony and his or her own statements.
This is part of what is known as the exclusionary rule, and is largely how the fourth amendment to the US constitution is enforced, protecting citizens from the government, and helping to provide due process of law to citizens. It does not throw out all evidence - just evidence that was gathered as a result of illegally gathered evidence.
No system of justice is perfect, and America's is certainly no exception...but I do firmly believe in a certain level of protection from the government. When we talk about excluding evidence, often, it is not over a small mistake from the police (e.g. mishandling chain of evidence custody). Rather, it often involves flagrant violations of the civil rights that help define us as a people. As others have suggested, how else would we incentivize the government to follow the rules of the road (or disincentivize it from breaking its own laws)?
This case is a simple argument of what constitutes probable cause to issue a legal warrant for a search. When arguing constitutional law, interpretation is pretty important...otherwise, the Supreme Court would be pretty bored. Personally, I think that the question has merit. This video was made by a person who claims that his car was illegally searched without consent based on the behavior of a drug sniffing dog. It seems reasonable that the result may be a false positive, triggered by the searching officer. If drug dog behavior is used for probable cause for searches based upon subjective, abusive (or easily abused) results, why shouldn't these be scrutinized by our highest legal authority?
IANAL