First the AG ("alien gospel") offers an extremely thin concept of sin. Sin is seen as merely breaking the rules for which you need forgiveness. There is no hint of sin as the deep and settled bent of the heart towards self-salvation and idolatry. Because the AG's account of sin is so shallow, listeners do not get the sense either that their sin is deeply unfair, wrong, and offensive to God or that it is profoundly destructive of their own lives. Instead, this view of sin as "rule breaking" leads listeners to see that their only problem is the legal consequences of the sin they face from the Divine Enforcer. Nothing in this presentation shows sin as intrinsically wrong, hateful, destructive, and shameful in itself.
As a result of this thin view of sin, the AG does not really clarify the classic gospel distinction between grace and works, between faith in Christ's saving work and faith in our own saving work. The average hearer of the AG will see themselves as saved, not primarily because of Jesus' death on the cross, but because they are sincerely submitting to God and begging for mercy and resolving to live a better life. Essentially, they do not see themselves as moving from faith in their own moral efforts (whether as secular or religious persons) to faith and rest in Christ's saving work. Rather, they see themselves as moving from living bad lives to living better ones. Their sins are forgiven, and God accepts them because they are now living for Jesus — not the other way around. [Tim Keller, Center Church: Doing Balanced, Gospel-Centered Ministry in Your City (Grand Rapids, MI: Zondervan, 2012), 269]
Further into his book, Keller attempts to defend his choice of defining sin as idolatry instead of the breaking of God's law. To this effect, Keller claims that defining sin as the breaking of God's law (1) is a shallow account of sin which does not show that sin is "unfair, wrong and offensive to God," (2) make it seem like it's only about the legal consequences of their sins, (3) result in basing one's salvation upon better rule-keeping, that is, works. Therefore, defining sin as the breaking of God's law is ineffective and will result in misunderstanding.
In response, one wonders what kind of a "traditional" presentation of sin Keller has in mind. A true presentation of Law and Gospel would clearly show that one cannot stop breaking God's Law, because God's Law demands perfection. It will also include communication of the wrath of God against sin, and that the grace of God in salvation is given as a free gift to those who cannot hope to merit it, so objections 1 and 3 are invalid. If Keller has in mind a simple Gospel presentation attempt like the "Four Spiritual Laws," then that probably applies, but just because some presentations of the Gospel which define sin as law-breaking are deficient does not imply that all presentations which define sin as law-breaking are deficient.
Now, if Keller is reacting to the simplistic and truncated law-breaking definition of sin, then yes, these problems would hold true. But his proposed solution is not the right one, for sin is now made subjective and the autonomy of Man not really challenged. The right solution is the solution that builds upon the biblical cosmology that begins with creation. Sin is law-breaking that is true, but in its fuller sense, sin is cosmic treason. Sin is saying to the Creator God who has created everything in this world including you, "No, I will not live as you have designed me to live." It is mutiny against the King of Kings, it is a denial of the Creatorship and Kingship of God. It is a denial of one's created status and the obligations that flow from that. That is the larger definition of sin, which is applicable to all peoples since all peoples are created and sustained by God. That is why the doctrine of creation is so important, and why Keller's attempt to relegate the creation/evolution debate to secondary status is a great error, for it results in Keller and those who follow him being unable to do full justice to the Gospel.