What some teetotaling Protestant groups want to believe is that Jesus drank only “new wine”, i.e, freshly squeezed juice. I even knew one who argued that Jesus would do the water-to-wine miracle whenever necessary to avoid the alcohol. He was young and enthusiastic, and hadn’t thought things through very well…
As far as proving the argument by timing the harvest, we have only to look at the growing seasons that exist today in the Holy Land, taking note of which crops ripen when, and compare them to the growing seasons as described in the Old Testament. The seasons are the same.
The Jews had harvest festivals throughout the year, and at each festival were to offer a specific type of “first fruit” of that particular harvest. See Leviticus Chapter 23. (My Protestant NIV Study Bible has a wonderful chart that summarizes all the feasts with their scripture references. God, I LOVE using my Protestant NIV Bible to prove Catholic teaching.

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So the fact that Jesus had to use fermented wine at the Last Supper is very easy to prove.