Let me break this down by looking at the text. For ease, my comments are in blue with the text presented in black.
4 Now there are varieties of gifts, but the same Spirit; 5 and there are varieties of service, but the same Lord; 6 and there are varieties of activities, but it is the same God who empowers them all in everyone. 7 To each is given the manifestation of the Spirit for the common good. 8 For to one is given through the Spirit the utterance of wisdom, and to another the utterance of knowledge according to the same Spirit, 9 to another faith by the same Spirit, to another gifts of healing by the one Spirit, 10 to another the working of miracles, to another prophecy, to another the ability to distinguish between spirits, to another various kinds of tongues, to another the interpretation of tongues. 11 All these are empowered by one and the same Spirit, who apportions to each one individually as he wills.
Here, Paul states gifts are given in variety, with different believers receiving different ones. Among the ones he lists, is faith. There is no indication that these gifts are common to all, but that they are distinctly given to individuals that each is forced to rely on the other. The passage begins with the idea that these gifts are given individually and in diversity, and it ends with the idea that they are individual. There is no indication that any of them are common to all.
12 For just as the body is one and has many members, and all the members of the body, though many, are one body, so it is with Christ. 13 For in one Spirit we were all baptized into one body—Jews or Greeks, slaves or free—and all were made to drink of one Spirit.
14 For the body does not consist of one member but of many. 15 If the foot should say, “Because I am not a hand, I do not belong to the body,” that would not make it any less a part of the body. 16 And if the ear should say, “Because I am not an eye, I do not belong to the body,” that would not make it any less a part of the body. 17 If the whole body were an eye, where would be the sense of hearing? If the whole body were an ear, where would be the sense of smell? 18 But as it is, God arranged the members in the body, each one of them, as he chose. 19 If all were a single member, where would the body be? 20 As it is, there are many parts, yet one body.
21 The eye cannot say to the hand, “I have no need of you,” nor again the head to the feet, “I have no need of you.” 22 On the contrary, the parts of the body that seem to be weaker are indispensable, 23 and on those parts of the body that we think less honorable we bestow the greater honor, and our unpresentable parts are treated with greater modesty, 24 which our more presentable parts do not require. But God has so composed the body, giving greater honor to the part that lacked it, 25 that there may be no division in the body, but that the members may have the same care for one another. 26 If one member suffers, all suffer together; if one member is honored, all rejoice together.
Here Paul makes his argument by analogy speaking of the functions and aptitudes of the body, especially highlighting that there is a special honor reserved for the parts often seen to have the least honorable function. The need for diversity is especially central and made clear by imagining if the body was entirely composed of one part. Again, the central idea is unified purpose through diverse endowment.
27 Now you are the body of Christ and individually members of it. 28 And God has appointed in the church first apostles, second prophets, third teachers, then miracles, then gifts of healing, helping, administrating, and various kinds of tongues.
The functions within the church are made explicit, each requiring its own gifting.
Are all apostles? Are all prophets? Are all teachers? Do all work miracles? 30 Do all possess gifts of healing? Do all speak with tongues? Do all interpret?
Here Paul asks hypothetical questions, of which the common answer is implied to be "no." Now, he does not explicitly mention faith but as it was expressly identified alongside the other offices and giftings mentioned it does not seem it would be an inappropriate shift to include it.
But earnestly desire the higher gifts.
And I will show you a still more excellent way.
Now Paul encourages desiring gifts and bridges the thought with what is to come, priming the reader to keep what has been said up to here in mind as he highlights a better focus than simply desiring gifts for the gifts sake. This thought links his discussion of diversity in the body with the expounding on the supremacy of love in 1 Cor 13, allowing us to discern his purpose in focusing on love.