Point 1 doesn't logically follow from the Miranda rights. In fact, that's the whole point of the adversarial court system: everything you say will be used for and against you, by the defense and the prosecution respectively.
The prosecution has no incentive or motivation to present evidence that could be used to the benefit of the defense.
https://en.wikipedia.org/wiki/Discovery_(law)
Specifically:
>Under the rule set forth in Brady v. Maryland, the prosecutor is obligated to provide to the defendant any information that is exculpatory or potentially exculpatory, without any request by the defense. Further discovery is available if initiated by the defendant. For example, a discovery demand might be for production of the names of witnesses, witness statements, information about evidence, a request for opportunity to inspect tangible evidence, and for any reports prepared by expert witnesses who will testify at trial.