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Barely any different from OP's aside from that I couldn't give a toss about auxiliary features. The only one that I can recall particularly enjoying is the Battle Factory in Emerald. PWT seemed decent but I never actually did anything more than the compulsory tournament
Audio is important to me because the music provides at least half the atmosphere for a location
Visuals more in the sense of art design than graphics
Last time I did any multiplayer you still needed a link cable, & even then it was more about trading than battles (although I did co-op Battle Tower once or twice & that was fun, even if we were shit)