I really think battles in Contests are fine. I agree not every competition in the show needs to include them, but they're the easiest way to keep them interesting, at least consistently. They just naturally lend themselves to more creativity, plus they're the easiest way to show off multiple members of a team (unlike, say, a race, which would only be possible for fast mons). And I still think the show was actually really good with addressing the vocal minority's complaints with them, at least by DP.
Plus I think some people are missing that Pokemon do need to be strong for any serious physical activity. I mean they're superpowered monsters, it kind of comes with the territory? One moment that sticks out to me was Kenny's performance in the Grand Festival: he underestimated how strong Empoleon's Flash Cannon was, which ruined his combo and got him eliminated. Strength is almost always a thing, even in competitions that act like it isn't.
Anyway, besides that and all the typical battle goals, I also voted for the Club tournaments. They're definitely not "goal" territory but they were a fun way to get the whole cast interacting, really wish they hadn't ended at BW. Things like races and Poke Ringers are fun one-offs, and I liked the episodes with them, but they'd probably get old if they appeared too often.
Showcases are the ones I dislike conceptually though. I guess I can respect a non-battling competition in theory, but the execution was so botched I really can't give them much credit for that.