I'm asking if that quirk of orthography was possibly aided due to an alternate pronunciation of 'y' that reinforced it.
No, there was no such alternate pronunciation.
At the time that thee was still in heavy use in much of the English-speaking world, it was as different to the ears and eyes of English speakers as we and I are today in terms of the perceived meaning, and as different as "thought" and "yacht" in terms of phonetics. Thee did not merge with ye or thou with you, but rather in most regions people simply stopped saying the singular thee and thou except perhaps when they came across it in their prayer book. And when they did say it as part of the Lord's Prayer, they were not confused as to how it was pronounced. Thee just died a death out of politeness. People were using the polite plural for more and more groups of people, to the point where nobody was left being covered by the more familiar singular. (Notably, when Quaker "plain speech" revived thee with a more egalitarian use, the perceived impoliteness led some to beating up said Quakers).
Meanwhile, the spelling had already become thee in almost all use, and when printing started in the English language Þ had already largely been abandoned. Caxton did include Þ in his sorts, but made little use of it. By the time English was being printed, thee was not spelled with a thorn.
The main exception the lack of the letter Þ in English printing is as part of the scribal Þe, which of course brings us to the other part of your question, but the inclusion of Þe was akin to the inclusion of & and likewise that scribes at the time still used Þe for the was pretty much exactly analogous to people using & for and today.
That Þe became Ye is due only to Þ looking more like Y in the sort of blackletter typefaces used (where the upper part of the bow of the Þ is thinner than the rest) than any other letter. Had the print styles at the time been closer to the sort of serif and sans-serif typefaces we use today it would likely have become Pe instead, and people would be saying "Pee Oldie shoppie".