French mothers and Irish mothers were both mothers? (nyuck, nyuck, nyuck.) I think the roots of the words "Gaelic" and "Gallic" are different. "Gallic" has a Latin root and "Gaelic" has a Gaelic root.
I think you are wrong about Gallic having a Latin root. It comes from the name of the Gauls, the Keltic people who inhabited what is now France and northern Italy. The Romans called them Gauls, but that was the name the Keltic tribes in this territory gave themselves collectively, IMO, as can be seen by the fact that when one of these tribes took off on a raid in Greece and continued on into Asia Minor (Turkey today), they created a small country whose name has persisted until today as Galatia, a people whom Paul sent an Epistle to. See post 13681 above.