In my very limited and new experience with saddle fitters, they come to you with a bunch of saddles, and you have as many saddles as possible available for them to try, and you try them all on the horse and evaluate fit. Where the rubber really meets the road in this enterprise is with saddles that have flocking ports so the flocking can be adjusted right then and there.
Particularly if this is a young horse or a horse just starting in work, their back will change as they age/get fitter, and the ability to change the flocking as the horse changes is critical.
Knowing what I know now, I would never buy a saddle that didn’t have flocking ports and could be adjusted easily.
The saddle fitter I am most familar with is a representative for a high end saddle maker; but she happily evaluates and adjusts fit on other brands of saddle. So while I understand the recommendation for an independent fitter, some maker reps are more interested in what works for you and the horse.