The reference in the 5th Amendment applies only to the federal government and its courts and agencies. The reference in the 14th Amendment extends protection of due process to all state governments, agencies, and courts.Constitutional Topic: Due Process
It is true that there was a debate about who was affected by the Bill of Rights, but it is not true that it "was considered to only be [a] limit on the power and authority of the Federal Government." If I may rearrange the words so that it is more accurate, "Many people considered it to only be a limit on the power and authority of the Federal Government, so Congress acted to close any argument on the matter by the state and federal courts, as well as the state legislatures, not necessarily because they agreed with the assessment."