The name "hornpipe" derives from a windcap double-reed instrument known as the "pibgorn" in Wales and "stock-and-horn" in England and Scotland. It has a limited range, comparable to a bagpipe chanter, and can't play most of the tunes we now call hornpipes. It *could* play the earliest examples of the genre, which were usually in 3/2 and seem to have originated in northern England. (These tunes are older than 9/8 jigs and seem to me to be the ancestors of them).

As far as I know the Trumpet Hornpipe was first published by Kerr in Glasgow in the 1880s. It would probably be a bugger on a trumpet but doable as a showpiece on the cornopean or cornet (hugely popular at the time the tune was first printed). The repeated notes at the start are a typical feature of brass music.