Earlier quoted context omitted.
Where are you getting this information? 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.
The digamma was never used in Homer's written form, but the poetic metre in some place is done such that there would be a digamma there and in some places, they don't modify the metre to adapt to the digamma.
When the The Odyssey was composed, widespread use of digamma sound was still a big thing.
I can easily believe that anachronistic language was added in the process of writing them down, but the original audience wasn't people reading in a book; it was people who were listening to a poem that had never been written down.
There is a huge amount of anacronistic content, but when it was first performed, language was relatively contemporary at the start.