I think the author missed an opportunity to point out what I feel is the the most convincing argument against labeling these operations as "Effectively Constant". The only reason we drop constant factors out of Big-O expressions is because we measure as N approaches infinity. Once you start reasoning about the size of N being bounded, the rationale for dropping constant factors vanishes.
The cost of maintaining a 32-element tree node might not be measurable as N approaches infinity, but for N < 2 ^ 32 it is, especially in relation to the cost of maintaining 2-element tree nodes. Once you consider constants, the runtimes are likely comparable between a binary tree with depth 32 and a 32-ary tree with depth 6 (with a skew towards the 32-ary tree for locality of access).
The cost of maintaining a 32-element tree node might not be measurable as N approaches infinity, but for N < 2 ^ 32 it is, especially in relation to the cost of maintaining 2-element tree nodes. Once you consider constants, the runtimes are likely comparable between a binary tree with depth 32 and a 32-ary tree with depth 6 (with a skew towards the 32-ary tree for locality of access).