Are there any known comparison sorting algorithms that do not reduce to sorting networks, such that each element is compared $O(\log n)$ times?
As far as I know, the only way to sort with $O(\log n)$ comparison on each element is to construct an AKS sorting network for $n$ inputs, and run the input on the sorting network.
AKS is not easy to implement and has an impractical constant factor, so there are motivations to search for other algorithms.
An algorithm with $O(\log^2 n)$ comparisons per item which does not seem to imply a sorting network is presented here. (iirc, this was first presented by Rob Johnson at Stony Brook's algorithm seminar).