“However, Pashuteh Yid’s question refers to Hagbah and Glilah, when the Torah is already wrapped up and a person catching it would touch only the velvet covering and not the actual parchment.”
the Sefer Torah is not wrapped up during Habboh and Glilah. it is exposed, that was his question.
it is not wrapped up until after Hagboh and Glilah are completed.