Re the harp: It wasn't that traditional tunings didn't allow accidentals; it was due to the diatonic nature of the Irish harp, which in those days had no semitone devices. (Pedal harps did, but they flourished in a different environment.)

On Celtic harps, if they were accompanying a song or another instrument the extra sharps and flats could be dealt with by simply leaving the 3rd out of the chord and letting the singer or other musician give the melody while the harp supplied surrounding harmony. Or, the harper could tune a particular note as a natural in one octave, and its sharp or flat in another octave. (Old tricks, but they still work!)

Incidentally, the gathering of harpers at Belfast was in 1792.