Rather, Celts appear to have arrived in Ireland in small numbers between 600-300 B.C.E. and the combination of that culture with the indigenous one gave raise to Gaelic culture. The Goidelic substrate hypothesis even goes so far as to suggest that certain words in the Gaelic languages are from a pre-Gaelic Irish language.

Any substrate that might be in the Gaelic language is far less important or recognisable than the Celtic origin of the vast majority of both vocabulary and grammar. It doesn't tell us all that much about it given how limited the proposed vocabulary is. Norse and Norman have had far more influence than any proposed substrate. Any languages we know to have been spoken in ancient Ireland like Iarmbelre or cruithnis are also thought to have been Celtic.

Saying the Irish are Celtic because our ancestors assimilated Celtic culture and language is like saying the Irish are Anglo-Saxon because we do they same with British and American culture today.

Likewise, saying the Gaels weren't Celtic is like saying the English aren't Anglo-Saxon, the Portuguese aren't Latin and so on. You can't really compare modern nations to historical macro-linguistic groups.