Hello again. I'm going to leave the weird tendency to believe weird things about The Celts™ (which you identify, with dubious accuracy, as "popular belief") to one side. Once again, the Victorians have a lot to answer for.
Kari Sperring, alias Kari Maund, received an excellent degree, but appears to have abandoned sober academic argument for books with titles like Princess Nest of Wales: Seductress of the English about 20 years ago.
Yes, most medieval societies are strongly patriarchal. The same can be said of many modern societies. Again I refer you to Judith Bennett's concept of patriarchal equilibrium. It's also true that women in medieval societies were subject to various forms of legal disability. But their legal disability does not mean that they were "by and large, property," in economic, social, or cultural fact. Were their options limited in law? yes. Does this mean that they did not participate in farming, in crafts, in arts, in ruling, in the religious life? No, no, no, no, and very definitely no.
I'm also going to gently suggest that the desire to find women doing the most archetypically masculine activity as evidence of their opportunities is... limited and limiting. Since you use the terms "early," "ancient," and "medieval" Ireland interchangeably, I'm not sure if The Ulster Cycle will be useful to you.
Four Courts Press, Dublin is generally a good place to look for works on medieval Ireland:
This work focuses on ideas of gender and power in medieval Ireland:
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