Clearly some parts of the federal Constitution apply to the states. The question is "which ones?". The 14th amendment applies to the states by it's very terms. "no state shall" for example. What is it that "No state shall" do? Violate the "priveleges and immunities of Citizens of the United States", defining them as "All persons born or naturalized in the United States, and subject to the jurisdiction thereof" What are the "priveleges and immunities"? The amendment does not say, but it's author/sponsor Senator Bingham stated explicity, in March, 1871, that the Fourteenth Amendment was designed to apply the Bill of Rights to the states. Noting that the first eight amendments "chiefly defined" the privileges and immunities of U.S. citizenship, Bingham stated, "These eight articles I have shown never were limitations upon the power of the States, until made so by the fourteenth amendment."
As to why have state Constitutions if the federal one applies to the states, it had already been observed that the states are free to explicitly protect more rights than the federal Constitution explicity protects. They also have a need to detail those powers not prohibited to them by the federal Constitution. Most of the Federal Constitution is a blueprint for the operation of the federal government, laying out both it's organization and the powers of its 3 branches. Those parts obviously do not apply to the state governmetns, the people of the states are free to organize their government as they see fit, for the most part. For example Nebraska has a Unicameral legislature, a practice that other states should, IMHO, take up, not a separate House and Senate, as do the federal and the other state governments.
One thing that bothers me with tying that into the Bill of Rights is that it is implying that those Unalienable Rights outlined in the Bill of Rights are simply "priveleges" granted by the federal government.