With most of what you say I have no dispute. However, “As my Father had sent me, I send you” is not a throwaway phrase. With it Christ, one with the Father, reminds us of the Incarnation where the Father indeed sent His human person, and He says that He likewise sends His apostolic Church as human persons. This is indisputable gospel.
Baptism is in the name of the Father, the Son, and the Holy Ghost, but people effect the baptism, and upon baptism one enters the Church. This is also indisputable gospel: when people convinced by St. Peter ask what should they do, he answers: repent and be baptised everyone.
Hence the Church of men is in the “send” relationship to Christ, and the baptism is what the Church offers, her primary gift of membership.
It is really amazing that anyone disputes this. It is not an obscure doctrinal issue.