If the "assumption" is so obvious and near-universal, why does Singer go on pompously to announce it.

Possibly for the same sorts of reasons that Eliezer wrote this big long thing to "restore a naïve view of truth", or that Nick Bostrom wrote this big long thing to explain why death is bad ... namely, that people have come up with all kinds of non-obvious, idiosyncratic rationalizations to justify the status-quo of starvation, ignorance, and death; that these rationalizations have, over the centuries, become cached thoughts; and that therefore getting back to "obvious and near-universal" basics is both desirable and nontrivial.

It's worth remembering, especially for people who are interested in meta-ethics, that often you can skip the confusion and jump ahead to Rough Consensus and Running Code. On rare occasions this will come back to bite you, but usually not (and if it does, you can jump back to the beginning).

Every argument must start from some unargued (because commonly accepted) assumptions. This is both logically necessary (as with axioms) and crucial for brevity of the argument. Since many truths are entangled, explaining all assumptions would eventually lead to listing a lot of our knowledge from many domains, and including many other arguments.