Suppose a serious crime takes place and I witness it. However, I witness it because I’m committing some other nefarious, though much lesser, act in the vicinity. The police talk to me and somehow they infer that this is the case, even though I haven’t said anything incriminating and they have no direct evidence. They don’t really care about my crime, they just want my info so they say “Alright, here’s what we’ll do. We will arrest you, but ‘forget’ to properly Mirandize you. Anything you tell us we won’t be able to use against you.” Could they do this to, in effect, guarantee me immunity?
Based on my extensive knowledge* of US Law, I can tell you that the detectives will get a telling-off for incompetency, which would probably outweigh solving the case.
Also a defence attorney could claim to the jury that there was a conspiracy. You make up false testimony and they arrange to let you off a crime.