In the distant past, the functions of author, editor and page designer were separated. The author wrote things; the editor fixed errors (both objectively and subjectively) and selected choice bits to highlight; the page designer made things look the way they should. The editorial intermediary didn't consult the author much, and gave limited direction to the page designer.
Pull quotes appeared where the page designer put them.
Because the editor could not predict where the pull quote would be located on the page, it made sense that the quote would not actually be removed from the text body, but would instead be duplicated as a stylistic element and remain in context.
For most web publications, this no longer makes sense.
For most web publications, this no longer makes sense. The page size is malleable and the browser will adapt. If you use pull quotes, you can guarantee that they sit in the text flow replacing the original text, and you can't guarantee any other placement. So: don't repeat yourself.
Don't repeat yourself.
It breaks your reader's flow.