In 1968, Congress attempted to overturn Miranda for federal cases by enacting the Crimes Control And Safe Streets Act. The Supreme Court held that part of the statute unconstitutional in 2000.
Birch Bayh gave an interesting statement on this issue for the Senate Judiciary Committee Report on the bill. He began by noting that his Subcommittee had held extensive hearings on the issue. His takeaway was:
This entire area is an extremely difficult and complicated one. Law enforcement officials almost unanimously agree that Miranda did in varying degrees cause them considerable difficulty. On the other hand, there was strong evidence expressing reluctance to support an effort to rearrange, restrict or repeal the Fifth Amendment guarantees.
After stating that he thought the bill’s Miranda provision would be struck down, Senator Bayh added this thought:
However, I do believe that [the provision] could serve, as Senator McClellan has suggested[,] as an admonition to the Court, that strong sentiment and cause exists against the further extension of the doctrine pronounced in Miranda. In addition, it is hoped that the consideration of this matter by the Congress will cause all law enforcement agencies to reexamine the actual holding of the Court in Miranda. Much evidence was presented in the hearings of the Subcommittee on Constitutional Amendments to the effect that some well-meaning jurisdictions had extended the holding of Miranda to be more restrictive on the police than was actually the intention of the Court.
Bayh made a similar observation about the Warren Court’s school prayer cases. (In other words, they were being read too broadly.)