Radix sort's time complexity is O(kn) where n is the number of keys to be sorted and k is the key length. Similarly, the time complexity for the insert, delete, and lookup operations in a trie is O(k). However, assuming all elements are distinct, isn't k>=log(n)? If so, that would mean Radix sort's asymptotic time complexity is O(nlogn), equal to that of quicksort, and trie operations have a time complexity of O(logn), equal to that of a balanced binary search tree. Of course, the constant factors may differ significantly, but the asymptotic time complexities won't. Is this true, and if so, do radix sort and tries have other advantages over other algorithms and data structures?
Edit:
Quicksort and its competitors perform O(nlogn) comparisons; in the worst case each comparison will take O(k) time (keys differ only at last digit checked). Therefore, those algorithms take O(knlogn) time. By that same logic, balanced binary search tree operations take O(klogn) time.