I can't say I completly agree with this. Even though big-O notation is far from perfect, I assume analytic combinatronics has its problems as well: since it's based on the scientific method, it should have some errors (although I have not read about it, so I might be wrong).

Also, I never seen anyone claim that quick-sort is O(n^2). Usually it's considered only the average case, where it's also O(n log n). This is where I believe Analytic Combinatronics should come in: if you want to compare two algorithms with the same order of growth on the average case. Otherwise, I think it's better to use big-O (analytic combinatronics to compare linear vs. exponential growth algorithms seems a little bit of an "overkill").