For just as the body is one and has many members, and all the members of the body, though many, are one body, so it is with Christ.
 For by one Spirit we were all baptized into one body – Jews or Greeks, slaves or free – and all were made to drink of one Spirit.
My thoughts exactly.
I would like to expand on your idea, slightly.
The whole body is good, but not all parts of the body are interchangeable, indiscriminately. Not all parts of the body belong together directly.
Some parts of the body ought to be restricted to indirect service of other body parts.
For a hand is wonderful when holding another hand or brushing off an itch from the skin.
However, the hand is not good when it’s finger tries to go inside an eye.
Each part of the body has it’s proper place, and it’s proper relationship to every other body part.
This variation is also true (albeit in an all or nothing way) to marriage. Marriage is where two bodies become one flesh. Some people go together, and some ought never marry. Incompatible people need an intermediary.
What many people don’t realize is that two people can both be good and incompatible.
Each of us has our proper way to relate to every other member of the body. We need to focus on relationships that build each other up.
The point I’m making is that equality, taken too far, leads to chaos and injury.
Some relationships are simply toxic. We ought to avoid toxic relationships.
St. Paul is asserting the Unity of the Church: all are equal members of the Body. Converts, regardless of their origin and status, are all part of the One Body–there’s no division in Christ (see verse 27).
Now, if you are intimating that every member has an equal vote (as many non-Catholics promote) then the answer must be no. Since St. Paul asserts that God has placed a hierarchy in the Church:
28 And God has placed in the church first of all apostles, second prophets, third teachers, then miracles, then gifts of healing, of helping, of guidance, and of different kinds of tongues. (1 Corinthians 12:28)
Interestingly enough, this is one of “those” passages which non-Catholics avoid or fully reject.