'Ye Olde Shoppe' was never pronounced with a Y sound

Old and Middle English used a letter called thorn (Þ) to write the 'th' sound, as in 'the'. When printing presses arrived from continental Europe without a thorn character, printers substituted the visually similar letter Y, producing abbreviations like 'ye' for 'the'. It was always meant to be read and said as 'the' — the pseudo-archaic 'ye' pronunciation used in mock-historical shop names today is a much later misreading.

— Standard historical linguistics accounts, Thorn (letter) — Old English orthography through early modern printing