Interesting take.The scripture speaking of unwavering faith is directed at all who profess to be Christians / all attending church. This manner of preaching is very evident from Jesus, Paul and all others NT. They never differentiated harshly between those saved and those not. The classic example for me is 1 Cor 5:1 It is reported commonly that there is fornication among you (among you professing to be saved), and such fornication as is not so much as named among the Gentiles, that one should have his father's wife.
Did Paul really think these people were saved when he says things like 1 Cor 6:9-13? We know that Paul taught OSAS from Eph 1:13. So what we have in 1 Cor 5 and James 1:6-8 is evidence of speaking to people professing to be saved. Not those actually saved.
You're right that Paul did not delineate between the saved and unsaved, in that he did not judge who was saved and who wasn't. What he did do was call out false teachers preaching a different gospel, any that was in opposition to his.
I believe Paul knew well that the flesh warred with the Spirit in Christians just as he portrays in his own battles in Romans 7.
What our minds want, which is agreement with the Spirit, is opposed by what our bodies (carnal desires) want.
I'm sure Paul knew he was preaching and teaching to Christian believers and wannabe believers alike. Like he taught, he was building on the Foundation of Christ and you never knew when a new piece of "raw material" would be saved and added to the structure.