if we take it that Grimm's law is right about gw <-> w <-> q
qui/quo -> who
quad -> what
guerilla -> warrior
guerre -> war
guarantee -> warranty
we get that
Celtic, Gaelic, Walloon, Wales, Welsh, Cornwall, Gallic, all refer to the same tribe of people.
Obviously, the rooster association with France is because of the Latin stem gall- in gallus, meaning rooster, compare to the apparently unrelated pollo meaning chicken. It's possible that we have a p/q swap here, like penta/quint- (5), in which case gall-/poll- might be related (Q vs P gaelic, like welsh vs irish).
At any rate, this leads me to suggest that the Celts from France to Ireland originally called themselves *Gwal-, and my guess would be that it meant 'people', since that's more or less what other tribes call themselves, e.g. Bantu/Botho (people), Thiud/Deutsch/Dutch (people)... etc.
Obviously, there's a different tribe of Gaelic speaking people that take the name 'Brit' - as in Britain, Briton, Bretaigne, Brittany.
Some examples of Gaelic -> Latin:
righ -> rex -> German “recht” (as in ‘might is right’) - king
tarbh -> taurus (bull) … pronounced ‘tarv’, again we see Grimm’s law where v->u/w