There are two annual matchplay events, the other is held at Valderrama, Spain (used to be Wentworth, near me).
They are PGA world events, so they are not restricted to players holding cards on the PGA US or European tours respectively. They are open (assuming quailification) to PGA US, PGA European, and PGA Japan tour players.
There are two similarly organised strokeplay events in the US also, and probably some outside.
I guess money rules again. Let the worlds best players play in everything and you can charge big $$ for entry.
Ive seen the prices of some of the tickets Jerry has had for big events. When I went to the 1st Ryder Cup at the Belfry (1985?) I paid $75 for all 3 days. For next year, that same ticket approaches $2,000. (No I am not going even though it is close to me). Incidentally, I played Celtic Manor when it opened. I guess the trees will have grown a bit since then.