Why bother with comparison sorts?

Question

Algorithms like Timsort, Quicksort & Mergesort dominate the "real world" sorting methods. The case for these comparison sorts is quite practical — they've been shown to be the most performant, stable, multipurpose sorting algorithms in a wide variety of environments.

However, it seems like nearly everything that we would sort on a computer are countable / partially ordered. Numbers, characters, strings, even functions are amenable to some meaningful non-comparison sorting method. A candidate here is Radix sort. In general it will behave faster than O(n*log(n)), beating the theoretical comparison sort limit of n * log(n) by a wide margin in many cases with a complexity of O(K*n) -- K being the number of bits that are required to represent a particular item.

What gives?

Answer 1

Comparison sorts are based on a really nice abstraction: all you need is a way to compare two elements. Then, depending on your language, with templates (c++), interfaces (java), typeclasses (haskell), function objects (javascript) etc.. you can sort containers which can hold arbitrary types, the only thing you need is implement the comparison.

How would you implement Radix sort for arbitrary types? :)