Pecan trees are about as dirty as a tree can be! I never had your problem, but I did have an extention of my driveway put in and it went under a pecan tree. It was also very "white" concrete. What a mess. I'm not confident you could get the stain out with anything, much less anything you might want to use near a pool.

I'd be inclinded to experiment with additional staining of the flagstone to mottle the look of the total area. The idea bening you could do it once and as it was further stained over the future years you wouldn't have to "clean" it again and again. Rather extreme, but offered as a notion for your consideration. Alas, I have no ideas one how one might accomplish that kind of additional, mottled staining.

A variation on that I considered, but sold the house before I did it, was to try to make a concentrated "tea" of the shells and flooding the concrete and see if I could even tone it down so the stains would be less prominent.

David