'Ye' in 'Ye Olde ....' is a representation of 'the' and is pronounced 'the' and always has been. The 'y' that is used in print replaces the archaic 'thorn' character, which looked somewhat similar and stood for what we now write as 'th'. In fact the 'e' in our 'the' was written in those days as a superscript following the 'thorn' character.
'Ye' as in 'Ye Olde Shoppe' is the most well-known example and the point is actually about how it is pronounced. 'Ye,' except when it is the second person plural pronoun, means 'the' and is pronounced as we pronounce 'the'.
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