Wright first over-generalizes aspects of some atheists to all atheists. Atheists are people who, for whatever reason or cause, good or bad, do not believe that any God exists. There really isn't any consistent way to generalize much of anything else to all atheists; a lack of one single belief does not entail much else, because it's extremely difficult, probably impossible, to prove that any derived belief can derive from only one particular premise. It's not a big point, because Wright addresses one important variety of atheism, scientific naturalism, but over-generalization can lead to some severe errors.
Although he calls it "atheism," Wright attempts to describe what looks like scientific naturalism.
Atheism is a theological stance. It is a theory of theology, or, rather, of metaphysics which holds first, the that universe is explicable without recourse to any theory of god or gods; and holds, second, that human knowledge proffers no clear evidence of the nature of divine things, whether god is one or many, whether life ends in oblivion, reincarnation, or last judgment; and holds, third, the human conscience and human prudence is sufficient, without recourse to divine spokesmen, to instruct the conscience and human decency sufficient to motivate the will to follow the conscience; and holds, fourth, that no account is logically coherent of an omnipotent god powerless to remove evil from the world nor a benevolent god unwilling to do so; and atheism concludes from this and other reasons that there is no god, and that even if there were, we would owe him no love nor loyalty nor obedience.I say that Wright describes scientific naturalism because all of the theological or metaphysical premises he ascribes to atheism are conclusions of scientific naturalism. But it is important to note that these are conclusions, not metaphysical premises.
The actual premises of scientific naturalism are simply that we can obtain reliable knowledge about the world using logic and the evidence of our senses according to a specific method, the Scientific Method. Another way of stating this definition is that scientific naturalism metaphysically privileges the Scientific Method and calls its outcome "knowledge." The ontology of scientific naturalism is, therefore, the simplest description of how the world could be to explain and account for the evidence of our senses. Under scientific naturalism, the question of how the world actually is independent of our knowledge of the world is not a meaningful question. Underlying this view are two other metaphysical principles:* that specify the job that any candidate system of knowledge must perform. First a candidate epistemic system must be preference-independent; there must be a rigorous method to resolve any differences of opinion about what we know. Second, it must produce, as far as possible, a specific ontology, a description of the world compatible with only the actual evidence of our senses, not one compatible with a range of evidence. Because the Scientific Method does these jobs, it is a successful epistemic method.
*There are other metaphysical principles about what a candidate system of knowledge ought to do, but they are not immediately relevant.
It is important to understand the meaning of "supernatural" in the context of scientific naturalism. A "supernatural" proposition is not a proposition that specifies an entity or property "outside" the physical world. Instead, A supernatural proposition is a superficially truth-apt statement whose truth or falsity cannot be known. It's not even a proposition that cannot be known by the Scientific Method; it's a proposition that cannot be known by any system of knowledge that meets the meta-epistemic conditions specified by scientific naturalism. Scientific naturalism a priori excludes the "supernatural" only to the extent that we deny that we can have knowledge of a proposition that by definition cannot be known, which seems harmlessly tautological. Scientific naturalism does not automatically exclude any and all theories of knowledge alternative to the Scientific Method. It does exclude a priori those methods which fail to do the meta-epistemic job.
To the extent that the existence of a god can be known, that existence is, in Wright's terminology, a "policy" matter. Scientific naturalism is neutral on the actual existence on such a god: either we know that it exists or we know it does not exist; either way, we must adjust our ontology to fit our knowledge. But of course, according to the Scientific Method, no god exists. (Technically, there is insufficient evidence to reject the null hypothesis that no god exists.) To the extent that theists attempt to privilege alternative epistemic systems, none do the job specified by the meta-epistemology of scientific naturalism. To the extent that theists specify a god that cannot be known, it is incoherent to assert the existence or non-existence of "god"; the proposition "an unknowable god exists" is not even wrong; it's incoherent. How it fails depends on the particular definition, but as yet scientific naturalism fails to justify — and failure to justify is grounds for denial* — the existence of any non-trivial definition of god. All the premises Wright assigns to atheism, as is atheism itself, are conclusions of scientific naturalism.
*Keep in mind that under scientific naturalism, acceptance or rejection of any epistemic or ontological proposition is always provisional. We may reject a proposition one day, and, on additional evidence or more comprehensive theory, accept it the next day.
To the extent that theists frame the existence of a god in purely metaphysical terms, the issue can approach a "political" conflict. Much depends on how much common ground theistic metaphysics retains with the metaphysics of scientific naturalism. Most importantly, is there meta-epistemic common ground? If we disagree not just on a particular epistemic method, but on the job a method must do to be a legitimate epistemic method, then we cannot agree on what we know. It's a much more difficult job to budge scientific naturalism on meta-epistemology than on some particular epistemic method; denying the meta-epistemic criteria will place the denier in what seems to be an irreconcilable conflict with scientific naturalism. We're not, however, going to feel any shame or guilt that other people have irreconcilable conflicts with our metaphysical system.