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Lillie starts out as a timid girl who's spent much of her life so far under her mom's oppressive parenting (after Mohn disappeared), to the point that she's never chosen her own outfit and only wears the space jellyfish costume her mom forced her into. She displays a distaste for Pokemon battles because she doesn't like seeing Pokemon getting hurt. She's also on the run from Aether, and by extension Lusamine, after kidnapping the cosmic turd that was presumably being/about to be tortured in order to open Ultra Wormholes, which is the point where Lillie's distaste for seeing Pokemon getting hurt outweighs her subservience to her mom.
Once on the run, Lillie's still pretty defenseless, not wanting to be a Trainer because Pokemon will get hurt, and being under Kukui's protection can only do so much. Then you show up, all confident and willing to risk your life for the cosmic turd even without a Pokemon yet, and at this point Lillie begins to actually travel. She gradually warms up to the idea of battling despite her misgivings; she buys her own outfit and eventually changes into it after shit hits the fan; she gets inspired by your Trainer-ness and tries to emulate it (though not becoming a Trainer herself until the ending, when it no longer matters); she helps to turn the cosmic turd into a legend; and she gets the courage to tell her mom off.
That tl;dr said, there are certainly valid complaints about Lillie's role and progression. The gap between "Maybe battles aren't so bad" and "You know what, I can actually be a Trainer" is frustratingly large, for one, and I get that.
