Perhaps a slight misinterpretation. The passage about parts of the body being parts despite what they may think/decide, does not appear to have the purpose of saying that the parts are "interdependent" so much as it is to affirm that once saved in Christ, we are part of His Church from the common thread of the Holy Spirit/Jesus/God the Father being manifested in us compared to those who are not saved.
It is a stretch to say it lays the groundwork to make intercession by other flawed mortals a requirement for salvation.
If that was the polemic behind the "we gave you the Bible" assertion then there would be no contention, except as to their claim.