> 3. The input to a sorting algorithm like abacus sort is a list of numbers. Even if there are n numbers, the number of "beads" you need to simulate is exponentially proportional to the length of the longest single number.

I think this is where we'd mismatch: I'd assume baked into the problem space is that you have N numbers to sort, they are each of size K where K is fixed and bounded. As a solution to "sort these N 16-bit ints, for some arbitrary N". It would be same problem space that can be solved in O(n) with bucket/counting sorts.