Stable sorting algorithms maintain the relative order of records with equal keys
Some applications like having that kind of stability, most don't care, for examples Google is your friend.
As for you assertion that "folks use sorting mechanisms like Merge sort or Quick sort" I would bet that most folks use whatever is built into their language and don't think about the sorting algorithm all that much. Those that roll their own have probably not heard of heap sort (the last is personal experience).
The last and biggest reason is that not everyone is going to want a sorted heap. Some people want the sorted list. If average Joe Programmer's boss says "sort this list", and Joe says "Here's this heap data structure you've never heard of, boss!", Joe's next performance review is not going to be so great.
