You still need to write each element to the target, which costs O(sqrt(N)), so the "strict" running time of radix sort would be O(N^1.5) in layered memory hierarchies. In flat memory hierarchies (no caching) as found in microcontrollers, it reduces back to O(N).
But radix sort doesn't write randomly to the output array like that. Yes, each element ends up in the right place, but through a series of steps which each have better locality of reference than random writes.
In this way, radix sort is usually faster than an "omniscient sort" that simply scans over the input writing each element to its final location in the output array.
