Do this, as often as you drink it, in remembrance of me.” For as often as you eat this bread and drink the cup, you proclaim the Lord’s death until he comes (1Cor. 11.25b-26 ESV).
Christians can become overly introspective during the Lord’s Supper because of Paul’s warning in the next verses about observing the rite: Whoever, therefore, eats the bread or drinks the cup of the Lord in an unworthy manner will be guilty concerning the body and blood of the Lord. Let a person examine himself, then, and so eat of the bread and drink of the cup (vss. 27-28 ESV). The “unworthy manner” speaks about those who get drunk as part of the observance. The “one goes hungry” in vs. 21 can’t be punished for the “unworthy manner” since they are not misrepresenting, but, merely not able to partake. The “hungry” might not be keeping the Lord’s Supper (through no fault of their own), but the “drunk” are “eating and drinking judgment:” For anyone who eats and drinks without discerning the body eats and drinks judgment on himself (1Cor. 11.29 ESV).
Here is a good explanatory article about the New Covenant by Jon Bloom. Although Jesus explicitly only mentions The New Covenant at the Last Supper, it doesn’t mean He didn’t refer to it many times in his sermons and parables in implicit ways. When Jesus tells His hearers that unless they become like children in Mt. 18.3 they cannot enter God’s Kingdom, He is speaking to the effect of what the New Covenant produces: a new life. When Jesus told Nicodemus- You must be born again (Jn. 3.7 ESV), He was referring to the renewing produced by the New Covenant where the person of the Holy Spirit indwells believers so that they would “know” Him as children. The New Covenant established a relationship where they would be His people and He would be their God (see Jer. 31. 31-34).
Other implicit references of Jesus were the parables of New Wine and New Cloth (Mt. 9.16-17, Mk. 2.21-22). The “unshrunk” (from agnaphos) should be rendered “new,” since, in Mk. 2.21 it is referred to “new” in the description of it. Most commentators recognize the two parables as speaking about the new life in Christ, and thus, The New Covenant.How Can God Forget My Sins? What We Remember at the Table — Desiring God