I think that what Paul is actually saying in 1 Corinthians 11:20 is that "The reason you assemble is not to eat the Lord's dinner but rather to ingest the sacred symbols he showed me".
The reason I say that is that this is the only place where the term "Lord's supper" is used in scripture so there is no reason to think it a name for something. He's saying "the food is not the point".
Adam Denoon's translation and analysis is in line with my own observation, of the infinitive, which I welcome because I have no credentials or expertise. This is from his answer elsewhere on this question:
The participle ("συνερχομένων") provides the answer you're looking
for. As a present active, it renders a simultaneous action with the
main verb ("is"):
[While] gathering together, it is not to eat the Lord's Supper.
The Greek does not bear out that the gathering together itself is "not
the Lord's Supper", but that their actions performed during that
gathering (infinitive: "to eat") is what merits the rebuttal.
So just reading his translation, which is quite literal (though I would change "while" to "when"), Paul is saying that they are not gathered to eat the Lord's dinner. Apparently to the Corinthians they were just there to fill their bellies:
KJV John 6:26 Jesus answered them and said, Verily, verily, I say unto you, Ye seek me, not because ye saw the miracles, but because ye did eat of the loaves, and were filled.
Paul says that it would be better not to come together:
KJV 1 Corinthians 11:17 Now in this that I declare unto you I praise you not, that ye come together not for the better, but for the worse.
He complains about the respect of persons in their assemblies:
KJV 1 Corinthians 11:19 For first of all, when ye come together in the church, I hear that there be divisions among you; and I partly believe it. For there must be also heresies among you, that they which are approved may be made manifest among you.
That verse 19 is not about Paul praising the Corinthians for good order. This is plain because he starts out the verse with "for first of all,..." and when you have a "for" you need to find out what it is there for. In this case it points back to the verse saying "it would be better if you did not assemble".
The logistics are a nightmare:
KJV 1 Corinthians 11:21 For in eating every one taketh before other his own supper: and one is hungry, and another is drunken.
He says that he absolutely does not approve of their novel idea of eating in the assembly and cites more problems:
KJV 1 Corinthians 11:22 What? have ye not houses to eat and to drink in? or despise ye the church of God, and shame them that have not? What shall I say to you? shall I praise you in this? I praise you not.
The reference to "praise you not" harps back to verse 1 where Paul praises the Corinthians for keeping the customs that HE HAS PASSED ON TO THEM. But in the next two section he deals with 2 customs that deviated from what he taught:
- head coverings
- coming together in the assembly to eat
For head coverings he tells them to tell anyone who pushes this custom that "we don't have such a custom" or "we have no such custom".
For coming together to eat in the assembly he says "bad idea" "eat at home" but at the end he says "if you decide to come together to eat, then wait for each other".
Verse 22 leads me to believe that people bringing their own suppers was at least part of the problem because some had plenty and some had none.