My Toupet (partial Nissen) was done by a thoracic surgeon in a big university center. He's done many, many thousands, and my GI doc said he has the best outcomes in the area - which was the bottom line for me.
I'm not sure if they even do Hill Repairs here anymore. The doc and his resident staff do mostly forms of Nissen, although they were in the forefront of a lot of the other types as they were invented. Nissens here run in the 90%+ range of successes and they do a lot of cancer ops.
My op was for a giant paraesophageal hernia - why I wanted the best surgeon available. These can twist and then you have to be operated within 90 minutes. I'm an hour's drive from the center.
Here the ops are done by thoracic surgeons (usually called cardiothoracic surgeons although they don't have anything to do with hearts). A good bit of the operation is in the chest, because a critical part is cutting loose and pulling doen the esophagus so it won't reherniate. I wouldn't want anyone with any less training!