Unless the reward of eternal life is understood as God simply being faithful to His own promises, rather than creating a type of legal obligation. God created the "legal" obligation when He said that He would not "impute" a man's sins to his account and would, instead, impute the righteousness of Christ through faith.
Even as David also describeth the blessedness of the man, unto whom God imputeth righteousness without works,
Saying, Blessed are they whose iniquities are forgiven, and whose sins are covered. Blessed is the man to whom the Lord will not impute sin.
Cometh this blessedness then upon the circumcision only, or upon the uncircumcision also? for we say that faith was reckoned to Abraham for righteousness. How was it then reckoned? when he was in circumcision, or in uncircumcision? Not in circumcision, but in uncircumcision. And he received the sign of circumcision, a seal of the righteousness of the faith which he had yet being uncircumcised: that he might be the father of all them that believe, though they be not circumcised; that righteousness might be imputed unto them also: And the father of circumcision to them who are not of the circumcision only, but who also walk in the steps of that faith of our father Abraham, which he had being yet uncircumcised. (Romans 4:6-12)
Eternal life is not a reward for works we do, but an act of grace and mercy by God. He gives to us eternal life as a gift and we receive that gift by faith. It is then that we learn how to walk in righteousness and holiness through the indwelling Holy Spirit - which genuine faith gives us. It is then that we can produce the kind of fruit that truly honors and glorifies God and not ourselves. We have been freed from the slavery to sin and its hold and can walk in newness of life through the new spirit nature.
For when you were slaves of sin, you were free in regard to righteousness. But what fruit were you getting at that time from the things of which you are now ashamed? For the end of those things is death. But now that you have been set free from sin and have become slaves of God, the fruit you get leads to sanctification and its end, eternal life. For the wages of sin is death, but the free gift of God is eternal life in Christ Jesus our Lord. (Romans 6:20-23)
God created the "legal" obligation God has never created any legal obligation from Him to mankind. That's a primary point of Paul's teaching in Romans 3 and 4.
Eternal life is not a reward for works we do
So you say. But Paul through the Holy Spirit says contrary:
6 For [God] will render to every man according to his works: 7 to those who by patience in well-doing seek for glory and honor and immortality, he will give eternal life[.]
God will render (reward) "according to works," and the stated reward (to those who are obedient and persevere) is "eternal life."
Yes, that is what the verse says.
The rest of your paragraph following this first clause is but so much opinion with nothing offered to substantiate it. Accordingly, I'm disregarding it.