I think you are defining good by a different measuring stick.
““Either make the tree good and its fruit good, or else make the tree bad and its fruit bad; for a tree is known by its fruit.”
Earlier in the sermon on the mount Jesus compares trees to people
“Even so, every good tree bears good fruit, but a bad tree bears bad fruit. A good tree cannot bear bad fruit, nor can a bad tree bear good fruit. Every tree that does not bear good fruit is cut down and thrown into the fire. Therefore by their fruits you will know them.”
This He said in speaking of false prophets
However this is not limited to false prophets but is inclusive of all men. All men can be known by their fruits.
In the same sermon He says
“For I say to you, that unless your righteousness exceeds the righteousness of the scribes and Pharisees, you will by no means enter the kingdom of heaven.”
These were Law/Torah abiding men and Jesus is saying they will not enter the kingdom.
In His illustration of the trees, they are equated to a bad tree with bad fruits. It’s only common sense that if they don’t make it into heaven their fruits are bad.
But why? Paul explains later in Romans
“but Israel, pursuing the law of righteousness, has not attained to the law of righteousness. Why? Because they did not seek it by faith, but as it were, by the works of the law. For they stumbled at that stumbling stone.”
What Apostle Paul is saying is they were trying to earn their way into Heaven through good works and all their good works to God are filthy because they are done in the flesh or led by the flesh.
“But we are all like an unclean thing, And all our righteousnesses are like filthy rags; We all fade as a leaf, And our iniquities, like the wind, Have taken us away.”
And to explain further, those who through works attempt to attain righteousness are doing so being led by the flesh, self will, self determination, self ambition, self!
“Therefore, brethren, we are debtors—not to the flesh, to live according to the flesh. For if you live according to the flesh you will die; but if by the Spirit you put to death the deeds of the body, you will live. For as many as are led by the Spirit of God, these are sons of God.”
Now in this context of good, whereby good means to be led by the Spirit of God let’s reread the passage you brought forth.
“eternal life to those who by patient continuance being led by the Spirit (in doing good) seek for glory, honor, and immortality;”
It is therefore impossible that Apostle Paul intended to say that someone who is walking in the flesh can produce good fruits of the Spirit.
““For a good tree does not bear bad fruit, nor does a bad tree bear good fruit.”
Someone who is not born again, is not IN HIM and does not live by faith being led by the Spirit cannot do good as defined by Jesus.