There is a small book by Gustav Leonhardt from the 1960's in which he already argued quite convincingly that The Art Of The Fugue is indeed keyboard music, but some pieces need a third hand, which is not unusual considering the number of pieces for two players extant. In early music circles there is no longer any debate about this.
That said, it was common practice in the 17th and 18th centuries to transcribe multiple voiced keyboard music to chamber instrumentation and vice versa. There are exmples from Bach's own oeuvre and in his circle, and documents that he improvised these from the score. It was part of the learning program of his sons and students.
As for the public performance of any type of keyboard music in Bach's time, there is a rather new book by Siegbert Rampe on the contexts of keyboard playing which is on top of my buying list (it's in German, however). I had brief look into it at the library, there were organ and harpsichord recitals, but limited to private circles or special public events, like auditions for an organist's post, or the testing of a newly built organ. OTOH there are some collections of organ music that can only have been intended for use in public performance, like the Husumer Orgelbuch. Bach gave some organ recitals in his lifetime, but we do not know what he played, there probably was a lot of improvisation involved, as was common practice.