No, I don't think grace has any achilles heals. I think Calvinism has an achilles heal. And it has to do with the "T" in "TULIP" (i.e., the idea that man is prevented by nature from understanding that God offers salvation through the sacrifice of Christ and that their nature deprives them of any ability to go to Him that they may receive forgiveness of their sins).
No, that's not how any of this works. God reveals deep things about Himself to us by His Spirit (1 Cor 2:10), we receive His tesimony directly into our hearts where He lives (1 Cor 2:12), and our knowledge of the things of God increases (1 Cor 2:12). When we (His children) are in error on our doctrines, our loving Heavenly Father corrects us. Some people are more stubborn than others. But God does not give up until His correction is successful. These things are from Hebrews 12. So, if a Christian believes false doctrine, it is not because he wants his false doctrine to be true, it is because he is resisting the correction of the Holy Spirit to suppress the truth.
There are many verses that speak of God calling on people to repent, to turn to Christ, and be saved. Let's take the following passage...
26 Jesus answered them and said, “Most assuredly, I say to you, you seek Me, not because you saw the signs, but because you ate of the loaves and were filled. 27 Do not labor for the food which perishes, but for the food which endures to everlasting life, which the Son of Man will give you, because God the Father has set His seal on Him.”
28 Then they said to Him, “What shall we do, that we may work the works of God?”
29 Jesus answered and said to them, “This is the work of God, that you believe in Him whom He sent.” (Jn 6:26–29)
Jesus told them they should labor for the food that endures to everlasting life. They asked for clarification as to what work that was. Jesus answered that they should believe in Him. There is no mention or insinuation or assumption that the hearers were prevented by nature or by God's hand from believing in Jesus. On the contraty, God was giving them an invitation to put their trust in Christ and have everlasting life.
There are no Bible verses that say "our will is a slave of sin". All the verses that speak of us being slaves of sin are peaking of us, not of our wills. Besides that, your point that your will is not a slave of sin right now after having been saved does not hold water either because it is obvious that saved people continue to sin after salvation and can't stop sinning. Maybe you're just saying that we are liberated from our depravity just long enough to believe in Jesus and be saved and the rest comes later?
It is obviously written to those who are judgemental towards the lost and their rejection of God...
Therefore you are inexcusable, O man, whoever you are who judge, for in whatever you judge another you condemn yourself; for you who judge practice the same things. (Ro 2:1)