@D.W. what if at each step, there is not a fixed finite number of choices, rather some $k$ choices. In that case, there would be $O(k^t)$ different control-flow paths. By properly tuning, $k$ and $t$ , we can get $t^t \geq n$ or $t \geq W(n)$. Here $k$ is Lambert's W-function. @D.W. I think a little more explanation is needed why this cannot happen.
In my opinion, the logic is lying inside the register level. A processor should be able to map the entire range of $n$ different memory units (not necessary RAM units). Considering binary representation there is no way out to hold information of $n$ different locations lesser than $\log_2(n)$ bits. While we are talking in terms of an array, or considering hash-table can access data in $O(1)$ average time, we assume hash operation is performed in $O(1)$ time. In reality, for hash operation, it takes at least $O(\log_2(n))$ time (considering the processing of a bit or a constant number of bits at a time). This book is focusing a bit on these aspects.