Why doesn't Knuth's linear-time multiplication algorithm "count"?
While the algorithm you mention appears in Knuth's TAOCP, it is certainly not due to Knuth, and is more widely known as the Schönhage–Strassen algorithm; Knuth even attributes this algorithm to them in the text. This algorithm indeed runs in linear time in the so-called RAM machine, in which variables are allowed to hold integers of size $O(\log n)$, where $n$ is the size of the input. The bit complexity is, however, $O(n\log n\log\log n)$, and for this model, Fürer's algorithm is faster.

The quest for fast integer multiplication algorithms in the literature has concentrated on bit complexity as the complexity measure; this is like allowing your registers to hold only one bit (or only $C$ bits for some arbitrary constant $C$). It is expected that integer multiplication takes $\Omega(n\log n)$, though it's not clear whether this should be tight, and no non-trivial lower bounds are known (so this $\Omega(n\log n)$ is just a conjecture).

For a discussion of the "correct" model for practical multiplication of large integers, check Fürer's recent paper. His conclusion is in favor of the "practical" Schönhage–Strassen algorithm (there are two of them, and the other one has better bit complexity but performs worse in practice; Fürer addresses this issue in the paper).