Interested to learn if there was a trend worth noting, I Googled Medicaid again (New York, county, share, Medicaid, property tax) and got strikingly higher results. The first time I searched, I got 8,100 hits. This time I got 21,100. Wow!
Then I noticed that there was a spelling error in the word "property" in the first post (since corrected). That error may have been in the first search as well, which would have accounted for the smaller though still very significant number.
We'll keep checking.