Gaelic preceding Brittonic in Ireland?

It is conventionally believed that Gaelic preceded Brittonic speech
in Ireland. But couldn't Brittonic have come first, Gaelic speakers
arriving later from Northeastern Iberia? From the web site Ireland's
History in Maps I found the following info, though I'm not sure how
true it is:

the Erainn would have included the Menapii, Bolgi/Builg/Belgae/Fir
Bolgs and the tribes in East Ulster, the ancestors of the later Dal
Riata and Dal Fiatach (aka Uliad) and also the tribes in Connacht-
the Ui Maine and the Conmaicne. It seems that these would have
originally been Brittonic-speakers from Gaul.

the Laigin originally coming from Armorica (thus Brittonic
speakers?), and the Domnann and Gaileoin also being of this region
(Leinster) , so these would appear to have originally been Brittonic
as well.

The Iverni, too, appear to have originally been Brittonic.

This would leave the Goidels (including the later Dal Rhiada-
the "Scots", including the Robogdii) as having been the bringers of
Gaelic speech, perhaps from NW SPain?

Thoughts?

-Michael