Off the top of my head, rides in Japan that I would say are better than DL's HM:
Sindbad's Storybook Voyage
Monsters, Inc. Ride and Go Seek
Amazing Adventures of Spider-Man
Journey to the Center of the Earth
20,000 Leagues Under the Sea
Splash Mountain (TDL)
Tower of Terror (DisneySea)
Indiana Jones Adventure: Temple of the Crystal Skull
You could also argue for the following EPCOT Center classics:
Horizons
Journey Into Imagination
World of Motion
Spaceship Earth
Universe of Energy (bigger, longer Dinorama than on the DLRRR)
Listen to the Land (original version. Odd choice maybe, but I could see the argument)
Maybe DLP's original Space Mountain, The Great Movie Ride (when it was new and special), Kilimanjaro Safaris, Phantom Manor, Mystic Manor, Shanghai's Pirates, etc.
As for what's better than pre-2006 Disneyland Pirates? It's a matter of opinion. Something like The American Adventure or Meet the World might leave a bigger impression, despite not being rides. Is what makes something a "complete experience" based on run time? Queue experience? Emotional response? I enjoy some things more than the big E-tickets, myself.
Of course, all thing pale in comparison to the soon to be closed Armageddon show at Disney Studios Paris. Truly in a class of its own.