Sorting (or any other O(N log N) operation) is only "basically a free operation" for small N, and if you're not doing it very often. A billion times log base two of a billion is roughly thirty billion. If an operation takes a billionth of a second, your "basically free operation" is taking thirty seconds. Also, you definitely want to avoid doing an O(N log N) operation more often than you have to. For example, doing

  for element of listB:
    listA.add(element)
    listA.sort()

is going to be much, much slower than

  listA.sort()
  for element of listB:
    listA.insertInSortedPosition(element)

since "insert in sorted position" should be O(log N) — just use binary search to find the correct position. The first algorithm is O(N^2 log N) while the second algorithm is O(N log N). For any non-negligible N, you definitely don't want to be running in polynomial time if you can avoid it! And can you guarantee your algorithm will never be run on non-negligible N?

I mean, yeah, sometimes you can. And in that case, "premature optimization is the root of all evil," sure. If you're just pulling a couple hundred product records out of a database and then sorting them in code, then yeah, an O(N log N) sort is "basically a free operation."

But sometimes you want to do more complicated things than that, and then it's not true. So overall I think it's pretty irresponsible to teach that sorting is basically free (without caveats). That's definitely not always true!

O(N) search is not fast.

Hash tables and trees are O(1) or O(lg N) which are vastly faster than O(N).

Imagine a database with billions of records. O(N) search on this database will absolutely murder performance.

I am also an amateur in these things. If indexing is O(N), does that mean it has the same performance as a linked list? Basically that would mean you have to go through each and every item until you reach the index you want.

Does homomorphic encryption allow running operations blindly on an encrypted dataset and return encrypted results without ever decrypting the dataset?