My handlebars are slightly higher than my saddle but my position is neutral, not too high or low.

I once bought a bike from a shop where they were good at sizing bikes but not fitting bikes. They followed your shops philosophy and put my handlebars lower than my saddle. When I rode in the drops my thighs hit my rib cage. A good fitter from a different shop determined my handlebars on this bike were 4cm too low. It had nothing to do with my flexibility, or my type of riding, but more to do with short arms. I ended up buying a custom frame with a higher than average head tube to accommodate my short arms.

I agree with the others, find a good fitter. Check out their fitting credentials, what type of training, how long they have been doing fittings, how much do they charge, etc. Some shops charge a lot but reduce the cost if you buy a certain level bike from them.

Many things can be changed to dial in your fit, but, nothing can be done to make a poor fitting frame fit well.