On the contrary, Paul is quite clear that "doing good" is indeed connected to final salvation:
God “will repay each person according to what they have done.â€[a] 7 To those who by persistence in doing good seek glory, honor and immortality, he will give eternal life.
I am always fascinated to read the case against the plain reading of this text. People try the following:
1. Claim that Paul is speaking
hypothetically (there is no justification for this, pun intended);
2. Claim that Paul is describing a judgement
at which no Christian will appear (the text directly contradicts such a reading);
I politely suggest that no such maneuvers work -
we have to accept that Paul means what he says here and see how we can reconcile such a teaching with his other teachings about justification by faith.
And we can indeed reconcile these things: Paul's basic argument is that it is only those who are empowered by the Spirit who can do "saving good works". And on what basis is the Spirit given in first place?
Faith.