Best fix for holes in travertine/limestone/marble IMO is a grout that matches the color of the stone somewhat closely. Mix it up, squeeze it in with a rubber grout float, let it cure for a few minutes, wipe the excess off with a damp sponge. There are also epoxies used for this, but they're harder to work with, and generally used for a discreet void or two rather than extensive pitting/voids. With such a highly textured travertine like that it might also help to do a quick wet diamond grind/polish to it afterwards and expose more of the underlying material.
@MinerJason
Should I use sanded or non-sanded grout?
I'm sorry to derail, but I have 2 corners that have always made me worry, specially with jumping children. Do I have a way to round them out, as should have been done by the previous owner when he had the pool built? He had kids. Can they be ground down and given a nice curve perhaps?
I stumbled upon a product called EZ patch, made by a pool-finishing company in Arizona. I wonder if that will work, and if anyone has used that stuff.
Looks to me like they gave it somewhat of a curve. Yes, you can do more grinding to make the curve more sweeping, but how much you can curve it will depend on how curved the pool wall is underneath it., which isn't clear from the photos.
EZ patch makes all sorts of products, which one were you looking at specifically?