Note from J.Vaughan.
"Yet, if I am not mistaken, he uses an organ as harmonizing instrument [in Handel] for some of that recital disc, something for which, as I mentioned last time, I do not care."
This leads me to want to talk about Baroque texture. The point of departure for the Baroque is supposed to be the trio sonata. The texture consisted of a bass line to support the harmonies and two equal treble voices. These three voices were emphasized and composed according to the rules of counterpoint. This texture left a hole in the middle and might well result in empty chords every now and then, especially at the cadences. So a harpsichord filled in the harmonies based on the composed bass line and numbers appearing below the notes called a figured bass. Nowadays these parts are often composed (the composed notes are printed smaller), but in those days the harpsichordist extemporized, and the resulting part probably varied a lot according to the skill of the player. The revival of antique practice that occurred in the second half of the twentieth century probably means a return to extemporization.
The sound coming from the harpsichord drops off sharply immediately after the string is plucked. The harpsichord part in the trio sonata texture doesn't rise to the level of an actual voice, so it is possible to see how this balance might be affected if the part were played on an organ. The notes would sustain like the real voices and might come to rival them, changing and thickening the texture.
I am assuming that the quoted sentence refers to the organ used as the instrument for realizing the continuo [as the process of filling in the notes above the bass is called], something he doesn't say. I've seen this. Usually a portatif is used--a portable organ about the size of a large upright piano. It is difficult to picture a portatif in an opera orchestra pit. It would stick up and block the view. Theodora, the work in question, is an oratorio where realization with a portatif is frequently seen. No one cares if it sticks up.
The ideal texture of the classical era fills in the middle voices with composed parts, except in secco recitative which continues on as before. Mozart's arrangement for Messiah provides a full classical texture in the orchestra with extensive parts for French horn, both replacing the clarino trumpet and in the style of classical French horn parts. A natural horn player could still play high fast notes. The sound ideal had changed. A classical trio includes a piano, cello and violin. The keyboard is an equal voice.
Professor Gossett points out that the figured bass continued on into his era (Rossini, Bellini, Donizetti, Verdi) where it was realized on a piano. After a period, it would have become difficult to find a harpsichord. When I was a young person, the harpsichord was just beginning its comeback.
I don't think I have such strong opinions about it one way or the other. Handel was an organist.