Not at all. The sheep are sheep before the talk of works. How did they become sheep? By faith in Christ their Redeemer. Notice that the sheep ask the same question as the goats. "When did we see you...?" The sheep did their deeds because as Eph 2:10 says "10 For we are his workmanship, created in Christ Jesus unto good works, which God hath before ordained that we should walk in them." They had pleasing deeds because of their faith and did the things the Lord had ordained them to do.
The goats on the other hand are separated from the sheep as well before. Why? They weren't believers. They ask the same question of the Lord "when did we see you....?" Look at some prominent advocates of human care and what the world would call charitable. Are they all Christians? Have they done more than you as far as you know? Surely there is a reward for doing all those good things. The answer lies in the 'why' of it. The Lord speaks in the Bible of those who make a show of their works and notes that they have already received their reward: Matt ^:6 “Beware of practicing your righteousness before other people in order to be seen by them, for then you will have no reward from your Father who is in heaven. 2 “Thus, when you give to the needy, sound no trumpet before you, as the hypocrites do in the synagogues and in the streets, that they may be praised by others. Truly, I say to you, they have received their reward. 3 But when you give to the needy, do not let your left hand know what your right hand is doing, 4 so that your giving may be in secret. And your Father who sees in secret will reward you."
The goats may have many more 'good deeds' than you, but they didn't do them for God as they didn't believe in Him. Their works, when they don't call on the name of the Lord are as Isaiah said, filthy rags.
Are we all either among the sheep or the goats? I infer from the parable that all of the sheep did good works and I guess I can go along with your reasoning that their works only counted because they were done in the name of Jesus. But wouldn’t that disqualify a death-bed conversion (assuming you believe in that).
I know I ask a lot of questions but I do agree with the great Catholic convert John Henry Newman who said that a 1000 questions doesn’t make a doubt.
Scripture can be quite fun, can’t it? There are lots of mysteries and so little time.