Mousethief, I think Bruce isn't talking about the phoneme (sound) we spell with the character "y" (which does have a relationship to the sound we spell with "g") but about a character not used in modern orthography called a thorn, which when hand written sort of looks like a "closed y" and which represented the one of the two phonemes we spell as "th" today. The "th" sound is related to the "D" in modern germanic languages.

Thee and thou would have both been spelt with thorns. Like the long s, so often mistaken for an "f" by modern readers, it wasn't used universally, and the most similar letter was substituted as time went on.

Now -- you may be quite right about how "thee" is used grammatically! I've never looked at that closely. My closes familiarity with it is the difference between the Norwegian "De" and "du", which is a difference in the degree of formality rather than subjective/objective.