PGA is an organization. So it is the organization championship. They just happen to play most of their tournaments in the US. If there was another PGA, say in Thailand, then it may need the country designation.
I think its just another playing condition like tight, narrow fairways or tall rough. The best players can adjust their game to fit the condition. I get some people like a 2-1 pitchers duel in baseball and others want a 9-7 slugfest. I'm not a big hitter, I score with wedges and the putter so I like to watch a round that accentuates that.
I think it depends on whether they can also hold the green with long approach shots. DJ won the US Open on very fast greens. But he also put a 195 yard shot to 4 feet on the last hole. The Canadian Open this week is kind of the opposite. The greens aren't as fast as Oakmont, but hard as a rock. The real difference in US Open set ups are the rough and off fairway areas. Usually they are far more penal.
Augusta is fast, but players can hold the green on 15 with a long shot. Hard and fast would change that course.
I've played greens that are very fast for amateurs. Once I got use to them, it wasn't any different. Very slow greens can also create issues with putting.