That is the larger context of 1 Cor. 11, but in the passage in focus St. Paul speaks of "eating" and "drinking" next to "discerning the body" and is repeating the words of Christ "this is my body". So the "body" in that passage is no longer the Christian assembly but precisely the Eucharist being eaten.
The immediate section of I corinthians 11 are the Apostle's words Correcting Abuse of the Lord’s Supper
17 In the following directives I have no praise for you, for your meetings do more harm than good. 18 In the first place, I hear that when you come together as a church, there are divisions among you, and to some extent I believe it. 19 No doubt there have to be differences among you to show which of you have God’s approval. 20 So then, when you come together, it is not the Lord’s Supper you eat, 21 for when you are eating, some of you go ahead with your own private suppers. As a result, one person remains hungry and another gets drunk. 22 Don’t you have homes to eat and drink in? Or do you despise the church of God by humiliating those who have nothing? What shall I say to you? Shall I praise you? Certainly not in this matter!
23 For I received from the Lord what I also passed on to you: The Lord Jesus, on the night he was betrayed, took bread, 24 and when he had given thanks, he broke it and said, “This is my body, which is for you; do this in remembrance of me.” 25 In the same way, after supper he took the cup, saying, “This cup is the new covenant in my blood; do this, whenever you drink it, in remembrance of me.” 26 For whenever you eat this bread and drink this cup, you proclaim the Lord’s death until he comes.27 So then, whoever eats the bread or drinks the cup of the Lord in an unworthy manner will be guilty of sinning against the body and blood of the Lord. 28 Everyone ought to examine themselves before they eat of the bread and drink from the cup. 29 For those who eat and drink without discerning the body of Christ eat and drink judgment on themselves. 30 That is why many among you are weak and sick, and a number of you have fallen asleep. 31 But if we were more discerning with regard to ourselves, we would not come under such judgment. 32 Nevertheless, when we are judged in this way by the Lord, we are being disciplined so that we will not be finally condemned with the world.
33 So then, my brothers and sisters, when you gather to eat, you should all eat together. 34 Anyone who is hungry should eat something at home, so that when you meet together it may not result in judgment.
And when I come I will give further directions.
The context is the context. Paul is writing about relationships with fellow believers and how to approach the Lord's Supper. There are relationship considerations for every believer, for the failure to consider other believers in the context of the Lord's Supper is condemned by Paul. After identifying and condemning these practices, Paul reiterates what He received directly from the Lord - including a command to discern the Body. He then goes right back to abuses against other believers. Eating worthily involves a self-examination. In this context, a self-examination of treating every member of the Body of Christ with honor.
You vipers!