I don't think Homeric Greek was archaic at the time it was composed. It would have been by the time it was finally written down instead of being passed down orally.
There is a lot of archaism in the content, though much of it is fake archaism that doesn't reflect actual history.
Everything I can see puts that development well after the poems' composition around 800-700 BCE, though perhaps before they had been fully written down.
If I understand correctly, it's a minority but not completely fringe opinion among historians that a major motivation for inventing the Greek alphabet in the first place may have so that they would be able to write down Homer.
There is a lot of archaism in the content, though much of it is fake archaism that doesn't reflect actual history.