I'm not completely sure what you are saying, but do you actually think a heap sort is in general faster than a quicksort or mergesort? You realize that the worst case of a quick sort is easily avoided right? The only way it happens is if you have an already sorted array and you pick your pivots from the minimum or maximum values on every single partition.
It is this point that has much of the academic computer science community saying that no sorting algorithm based on comparing keys two at a time can beat heap sort.
I think you're the only one saying that. Where did you get this idea? I just showed you a quicksort being 10 times faster than a heapsort. You can try this for yourself.
Sure, we don't know how big n has to be. In practice, in an actual case, the n might have to be too big for current computers.
This is never going to be true. Quicksort and other sorts exploit locality much better. Partitioning an array gradually makes the sorting more local. That's going to work better relatively to a heap sort as data gets bigger, not worse.
Sure, in practice, for some value of n, on some list of n keys, some version of heap sort, and some version of quick sort, the quick sort might run 10 times faster in seconds of elapsed time than heap sort.
No, on any modern computer other sorts are going to beat a heapsort. It isn't exotic, it is how it works. Even if you have to scan every element first to get a distribution of your elements to choose your pivot, that is still going to be O(n log n) and it will still beat a heap sort.
Heap sorts are not the fastest way to sort. They can be consistent and they can split the time between inserting and extracting elements.