I think that the force of the argument that the Epistle of Clement is evidence of papal primacy is precisely that he was a bishop, not an apostle. It goes without saying that apostles had authority that transcended the boundaries of “diocesan” churches (to use an anachronism)–everybody in the early Church was bound by the authority of the apostles. But for a “diocesan” bishop to exercise such authority, that is noteworthy.
I don’t think the examples of St. James or the writer of the Epistle to the Hebrews have much impact here since there is no evidence that they were writing to anyone other than their own flocks. And both epistles are essentially hortatory.
And the epistles of St. Ignatius actually provide support for the argument on papal primacy, as has been pointed out already. Again they are by and large letters of exhortation, not letters of command like the Epistle of St. Clement. And the tone adopted by St. Ignatius with respect to the Church of Rome is very different, much more deferential and designating that latter church as that which “presides over [the] love”. As our Holy Father has said:
And precisely in the service of unity to the Catholic Church, the Christian community of Rome exercised a sort of primacy of love: “The Church which presides in the place of the region of the Romans, and which is worthy of God, worthy of honour, worthy of the highest happiness… and which presides over love, is named from Christ, and from the Father…” (Romans, Prologue).
GENERAL AUDIENCE, 14 March 2007]
I don’t think a huge amount of weight can be put on the Epistle of St. Clement taken by itself. But as has been said here already, it is definitely one valid piece in a full-orbed apologetic.