Are you aware that literally everyone will appear before the judgment seat of Christ to confess that Jesus is Lord and not just those in the body of Christ?
Romans 14:10 But why dost thou judge thy brother? or why dost thou set at nought thy brother? for we shall all stand before the judgment seat of Christ. 11 For it is written, As I live, saith the Lord, every knee shall bow to me, and every tongue shall confess to God. 12 So then every one of us shall give account of himself to God.
Philippians 2:9 Wherefore God also hath highly exalted him, and given him a name which is above every name: 10 That at the name of Jesus every knee should bow, of things in heaven, and things in earth, and things under the earth; 11 And that every tongue should confess that Jesus Christ is Lord, to the glory of God the Father.
So, the idea that the goats only represent unprofitable servants because they refer to Him as "Lord" the same as the sheep do is not a valid argument because literally all unbelievers, and not just unprofitable servants of Christ, will be confessing that He is Lord and will call Him Lord at the judgment.
As for how to identify the sheep, you seemed to indicate that anyone who helped feed and clothe the poor "would make them the sheep in that case". Remember, the sheep inherit eternal life in the kingdom of God. Does scripture say that we are saved and inherit eternal life by our works of helping the poor? There would be a lot of Christ rejecters and those who oppose God or don't believe in God inheriting eternal life in the kingdom of God if that was the case.
No, scripture says we are saved by grace through faith and not by works (Ephesians 2:8-9). So, you can't say that everyone who has helped the poor is automatically among the sheep on that basis alone.
Notice that Jesus said that anything that is done for the least of these His brethren is done for Him. So, that means He is differentiating between those who love others and help the needy out of their love for Him and desire to serve Him and those who either don't help the needy at all or who do so out of the wrong motivations (trying to make themselves look righteous or trying to earn salvation) instead of out of a humble desire to serve and obey Him.
We are saved by grace through faith and not by our own works of righteousness (Ephesians 2:8-9, Titus 3:5), but once we're saved God has good works prepared for us to do (Ephesians 2:10) which relate to loving others and helping the needy. So, what Matthew 25:31-46 is about is differentiating between those who are saved and belong to Christ and did the good works that God prepared for them and that reflect their faith and those who are not saved and therefore did not do the good works that God only prepares for those who are saved.