The flaw in your statement "But Paul is addressing the church" is that you assume everyone in the church at Corinth is actually a believer. We know that people in churches do, in fact, prove themselves to be otherwise. In Acts, Paul describes the "wolves" to the Ephesian elders--and those wolves come from inside the church, not from the outside. Anyone would be hard pressed to say the "wolves" are believers.
This is not true. "Carnal" is an old English translation of ψυχικός and ψυχικός means "unspiritual" or "worldly." It can mean "natural," but Paul's usage (four times in 1 Corinthians) shows that natural is meant only when talking about the body (as he discusses it 1 Cor 15).
The context of the pericope proves your argument to be incorrect. Here is the text:
[6] Yet among the mature we do impart wisdom, although it is not a wisdom of this age or of the rulers of this age, who are doomed to pass away. [7] But we impart a secret and hidden wisdom of God, which God decreed before the ages for our glory. [8] None of the rulers of this age understood this, for if they had, they would not have crucified the Lord of glory. [9] But, as it is written,
“What no eye has seen, nor ear heard,
nor the heart of man imagined,
what God has prepared for those who love him”—
[10] these things God has revealed to us through the Spirit. For the Spirit searches everything, even the depths of God. [11] For who knows a person's thoughts except the spirit of that person, which is in him? So also no one comprehends the thoughts of God except the Spirit of God. [12] Now we have received not the spirit of the world, but the Spirit who is from God, that we might understand the things freely given us by God. [13] And we impart this in words not taught by human wisdom but taught by the Spirit, interpreting spiritual truths to those who are spiritual.
[14] The natural person does not accept the things of the Spirit of God, for they are folly to him, and he is not able to understand them because they are spiritually discerned. [15] The spiritual person judges all things, but is himself to be judged by no one. [16] “For who has understood the mind of the Lord so as to instruct him?” But we have the mind of Christ. (1 Corinthians 2:6–16 ESV)
In v. 10, Paul says the wisdom of God is revealed through the Spirit. So, the things of God cannot be known without the inward illumination of the Holy Spirit. I doubt we'd disagree. Then, he contrasts the spirit of the world with the Spirit of God. His point is that believers have the Spirit of God with the express reason that we (believers) may understand the things of God. Paul goes on to say the transmission of the things of God happens not by human wisdom, but by the Spirit--"interpreting truths to those who are spiritual." Paul is clear here in his implication: If you do not have the Spirit of God, you have no real hope of understanding (or rightly understanding) the things of God. Then, what he made implicit in v. 13 he makes explicit in v. 14: The "natural" person does not--and cannot--accept the things of God.
So, Paul--here--is not talking about a believer; he is discussing an unbeliever. But, he is discussing believers and unbelievers in contrast--because of what is coming up in chapter 3. In chapter 2 he is clear: The "natural" man is an unbeliever because he "does not accept" and "is not able to understand" the things of God. Paul has already told us that we are able to understand the things of God if we have the Spirit. But with the natural man, there is no option. Understanding is not even available to the natural man, therefore he or she is an unbeliever.
Now, in chapter 3, Paul does equate the believers with being infants, not unbelievers. Here is the pertinent text:
[1] But I, brothers, could not address you as spiritual people, but as people of the flesh, as infants in Christ. [2] I fed you with milk, not solid food, for you were not ready for it. And even now you are not yet ready, [3] for you are still of the flesh. For while there is jealousy and strife among you, are you not of the flesh and behaving only in a human way? [4] For when one says, “I follow Paul,” and another, “I follow Apollos,” are you not being merely human? (1 Corinthians 3:1–4 ESV)
Notice that Paul is using "as," as in "I could not address you as spiritual people." In English when "like" or "as" is used it is called simile. It is a linguistic tool of comparison. So, Paul is comparing the "Brothers" (so, believers) in the church of Corinth as "people of the flesh" (so, non-believers). However... He does put the comparison defining clause "as infants in Christ" to define "people of the flesh." Now, we know from Paul's use of "brothers" that he is intending to address believers. What he says to them is important: They are certainly not acting like believers.
In other words, the Corinthian believers were fed "milk" by Paul when he was with them and they have not grown past being spiritual infants. It would seem Paul expected them to grow into "solid food," but they haven't done so. Instead of behaving like believers, they are behaving in a human way.
To conclude, then, Paul knows nothing of the so-called "carnal" Christian as described by so many (because the natural man does not and cannot understand the things of God). What he does in chapter 3 is to compare the believers to non-believers, which is to call into question their Christianity--though he does not discount their belief (calling them infants). The Corinthian believers are not showing the fruit of repentance and faith. Paul does not consider it proper or normal for Christians to remain "infants," probably because infant Christians look like unbelievers. The only way to know the difference is that infant Christians (because of the work of the Spirit) grow into mature Christians. And... only time will tell.
The Archangel