Capital of the US as in "capital city" would be more in reference of DC and the White house, invoking Executive Branch connotation. When you think the Capital of the US, you think DC and the White house.
Capitol as in Capitol Hill invokes the connotation of the Legislative Branch, or Congress specifically.
Both of these in regard to the Kavanaugh vote would seem to me as a passing off of the buck. As in its no longer in the Executives responsibility, we've done all we will do, its up to you Congress, and all eyes are on, and responsibility for success or failure will be directed at Congress. A slight hint as in, don't even dare think you will get away with last minute fuckery, you are being watched.