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.