Article VI - Yep. It does. "...shall be the supreme Law of the Land; and the Judges in every State shall be bound thereby...'
The vast bult of the Ratification sources are the newspaper articles and reports of campaign speeches. With the exception of Pennsylvania, Dean James Bond states, the State legislatures kept no records of their debates. The legislative debate in Pennsylvania, he comments, 'reads like a reprise of the six month campaign that preceded it'. There are a few of the Governor's messages, but 'most are quite general'. In the three states on which Bond concentrated, he observes that debate 'did not focus exclusively or even primarily on the first section of the 14th Amendment. The principal issue in those states was control of the national government'. Republicans feared that Democrats would wrest control of the House because with emancipation Southern representation would no longer be limited to three-fifths of the blacks as Article I(3) provided.
(snip)
Speaking in Chicago in August 1866, Senator Trumbull, who had piloted the Bill through the Senate, 'clearly and unhesitatingly declared of the Amendment to be 'a reiteration of the rights as set forth in the Civil Rightst Bill", which did not include any reference to the Bill of Rights. In Indiana, Senator Lane 'affirmed Trumbull's statement concerning the first section'; and Senator Sherman 'endorsed' those views in a speech on September 29,1866. Senator Poland spoke to the same effect in November 1866--Raoul Burger, The Fourteenth Amendment and the Bill of Rights, pp37-42
The Fourteenth Amendment and the Bill of Rights--PDF File
I would suggest the whole book. It is a PDF file but a good discussion of the Fourteenth and the Bill of Rights. BTW, Poland and Trumbull are two of the men sourced in the article you provided RKV. Seems they didn't mean exactly what your source would have us believe...