Boo, cop-out answer. The point of the question was to get you to stop thinking so close to in-game, all about weaknesses and moves, and think more logically. I find that the anime battling world and the manga battling world is a lot more interesting to debate about. Not because anything can happen, since there is some clear canon involved and you can assume a lot, except for those champions that barely got a mention. Like Steven for instance. Abilities do count a lot in battles, so I don't know where you're getting that from. Moves do virtually the same things, except more because they more depth to work with.
First of all, I didn't mean abilities didn't count. I probably worded it quite bad, but some abilities do in the anime what should be logical. Take Froslass for example, one of its abilities in-game is Snow Cloak (yes, I know I'm thinking about the games again, sorry) which raises its evasion in hail. When it starts snowing in the anime it is not more than logical Froslass won't be visible, same with Garchomp and Sand Veil and other Pokémon. Most abilities do get used, abilities like Run Away and others don't really count in my opinion because they aren't used in the anime and there's no need for them (as far as I know).
Moves do indeed get more depth, some don't and even lose some of their depth. The most famous example being Dig which can be used while surfing in the games and clearly cannot be used in the anime like that.
As for Garchomp vs. Paul I need to apologize again for my bad wording. I merely meant that Paul being fairly 'lower' than Cynthia managed to deal at least some damage to Garchomp. I know that Paul already traveled Kanto, Johto and Hoenn but Cynthia is the champion and Paul did mention he didn't win anything in those regions. Cynthia allowing Garchomp to take hits can be true, but one has to admit she didn't expect Paul to use his Torterra like that. She was genuinely surprised.
I agree that anime battling (don't know about the manga) is a lot more interesting to talk about than game battling. When I write out Pokémon battle scenes for stories I always use the anime, never the games, just because there is so much depth.
I would go in great detail and analyze every champion vs. champion battle in anime style if I didn't think it would take up too much space.
One thing I don't really get is how 'logical' would differ so much from thinking about 'moves and weaknesses'. To me weaknesses and moves mean Fire loses to Water due to Pokémon X using Water Gun on Pokémon Y. Logical simply means (to me) that a trainer would use Water Gun and not some Grass type move on a Fire Pokémon. Unless you mean to think about setting too. Like how you shouldn't be able to battle with a Fire Pokémon while surfing because he'd simply not be able to stand the water and faint instantly.
Sorry for apparently not answering the question. I hope this explained what I meant a bit, but if it didn't feel free to say so (or if I said something completely and horribly wrong)