Old English had the þ character (capital: Þ), used as a 'th' sound, and basically from old Norse influence (where now only Icelandic still uses it).
When printing came about, the thorn was already in decline, mostly used for a few common words(verify), so it didn't really make sense to add it and make things more complex.
But also, space was at a premium, so making common words like 'the' shorter was still a decent idea.
And since the thorn character written in decorative script looks a bunch like the letter Y, Y with an often-superscript e is what they used instead in printing, also using this pseudo-thorn for the few abbreviations of other common words starting with th, like this and that(verify).
So ye became an alternative printing for the word "the", understood as exactly the same thing.
These days, thorn is long gone, so when you put "Ye olde shop" on a sign, people will easily pronounce the y as the vowel (yee).
All this wouldn't be an issue, except ye ('the') at the time was entirely from ye, the plural pronoun (similar to modern american y'all).
So when you pronounce that shop's name with 'yee', it's basically neither the thorn, or the plural pronoun, because that'd be as gramatically incorrect as "Y'all shop", meaning you've sort of invented a third thing never really part of your language.