Today, I have data (and code) that confirms the distribution is sorting-algorithm-dependent.  For each sorting algorithm, 1,000,000 instances of the list {0,1,2,3,4} were sorted with a random comparison function and the relative frequencies (rounded to the nearest whole percent) of each number in each position were computed.

The distributions are significantly different among these sorts.  QuickSort appears to provide a uniform distribution.  I believe that this is because QuickSort will only compare a particular pair of elements once, whereas each of the other sorting algorithms may compare a given pair of elements more than once (and with a random comparison function, receive a different result from one time to the next).

Here is the Mathematica notebook I used to generate this data: Randomize by Sorting.nb.  As noted in the file, some of the code for the sorting algorithms was taken from other locations and may be/is subject to their copyrights and/or license terms (I reasonably believe that this use complies with their licenses and/or constitutes fair use.  Also, some algorithms exhibited improper behavior when trying to sort lists with duplicate entries using a normal comparison function as noted in the file, though this should have no effect on the data above.