I am not sure what your definition of "The Church" is.
I think that to the Orthodox, they think only they are "The Church".
It is my understanding that the church is the ekklesia - the gathering of the called out ones and it includes all redeemed and regenerated believers.
The church has both a universal aspect and a local aspect. The church is the Body of Christ. The gates of hell will not prevail against the church. Satan will and does attack the church, but he will never prevail. That has been promised and it is true.
As far as the eucharist, and the verses in your signature, those verses are closely followed by this one:
John 6:63 It is the Spirit who gives life; the flesh profits nothing; the words which I have spoken to you are spirit and are life.
In this verse, Jesus was telling the Jews that it was not His physical flesh which they would be given to eat, for the flesh profits nothing, but the Spirit who gives life, who is the Lord Himself in resurrection.
He is the real food and the real drink and we can take Him in in spirit any time and any place.
The bread and wine that we partake of in the table meeting is to be done in rememberance of Him, which we do once a week.