Re "Emmanuel" - the Companion to Hymns A&M says that while the 1853 source claimed to have got the tune from an old French missal, nobody had ever found it. Can the folks claiming an older origin actually point to it?

"The original Latin" is ambiguous. The hymn version is the one (in Latin) from 1710. It's a versification of the older antiphons, which don't fit the modern tune.

A different way to look for old tunes. Kodaly in "Folk Music of Hungary" identifies a number of tunes which are found among the Hungarians and also among their closest linguistic relatives, the Mari (a.k.a. Cheremis) of the Volga region. The two groups have not been in contact for about 1500 years, and nobody else anywhere sings the same tunes. So the obvious conclusion is that they predate the westward migration of the Hungarians.