Disclaimer: I don't own anything.
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Yellow finds it depressing that she doesn't have a clear goal in mind.
She constantly watches her friends, who each have their goals. There is Red, who wants to be the best trainer in the world (and maybe one day to take over the Viridian gym from Green). There is Blue, who wants more than anything to find her family and be reunited with them, and Silver, who shares her goal. There is Green, who also wants to be the best trainer and will continue battling with Red until he can defeat him, at which point he will continue training to ensure that Red doesn't beat him. And there is Chris, who dreams of being the best catcher in the world and surpassing her mother.
But when Yellow started asking herself what she wanted, she was surprised to find that she didn't have a clear answer. Yes, she would be thrilled if no Pokémon had to be hurt, especially with her empathic abilities. But even at her most optimistic, Yellow realizes that such a goal is impossible to attain. Trainers all over the world battle every day, and there was no way to ease the suffering of each Pokémon who had fallen in battle.
When she starts thinking about these things for a while, Yellow finds herself dangerously close to sympathizing with Lance, who had wanted to completely wipe out humanity besides a few chosen people to create a paradise for Pokémon. After her battle with the Elite Four member, she wandered the world for a while, glad that she could recreate conditions to ease the suffering of Pokémon. For a while, she was happy to see that the wild Pokémon she encountered seemed to be doing well. However, when she ran across two trainers battling on her way home to Viridian City, Yellow watched in a depressed fashion as one of the trainer's Pokémon brutalized the other so it wouldn't get back up.
At that point, she was brutally reminded that no matter what she could do for wild Pokémon, the trainers who commanded the Pokémon who battled could inflict more brutality on their Pokémon than they ever could via pollution or starvation. Yellow found it sadly ironic that the only to ensure that Pokémon didn't get hurt was to take Lance's path or something very close to it.
Yellow tried to find another goal for herself, she really did. She thought about volunteering at a Pokémon center – after all, if Pokémon had to get hurt, at least she could patch them up and make sure that they were as comfortable as possible. One week later, Yellow had to quit. It wasn't because she was extremely overworked and exhausted constantly – though that was part of it – nor was it even because of the bad memories of their trainers a few Pokémon had. For the most part, Pokémon had nothing but good to say, if only to ensure that the girl healing them didn't become upset. No, what caused Yellow to quit was the sheer apathy of many of the trainers who left their Pokémon for healing. The trainers would come, drop their Pokémon off for a while, and pick them up, only for the same trainers to come back in within hours with their Pokémon completely exhausted again.
The attitude that these trainers took towards training and battling with their Pokémon depressed Yellow. The trainers were so focused on their goals that they didn't bother to give their Pokémon any more of a well-deserved break than the time they could spend at the center. Although the Pokémon she healed tried to keep up a happy façade, anyone who watched them closely enough (or was able to read their minds like Yellow) would be able to tell that they were depressed over not being able to spend quality time with their trainers, instead of just training.
Worse, the Pokémon who were genuinely happy about their situation never came to the Pokémon center. The trainers of those particular Pokémon often took the opportunity to tend to their Pokémon themselves if they became injured, either to strengthen the bond between trainer and Pokémon or just to spend quality time with their friends. After a week of seeing the worse in people through their partners, Yellow just couldn't take it anymore.
She bemoaned the fact that it shouldn't even be necessary for Pokémon to battle each other. Many people took the opportunity to be friends with their Pokémon, and merely live with them – she had done so herself before her journey to help Red. However, it seemed like the bulk of the people and Pokémon she encountered were focused on beating each other up, through the excuse of battling. The one thing Yellow has never admitted to her friends was that at the end of the week she worked at the Pokémon center, she gave serious thought to what she called 'Pulling a Lance'.
At this point, Yellow was possibly the most unhappy she had ever been. She had just received first-hand proof that humans were as cruel and callous as Lance had claimed, and she didn't know what to do with herself. Her one wish that could be considered a goal – keeping Pokémon from having to be hurt – was impossible. Even if she was able to change circumstances so that wild Pokémon were happier, there was no way that she could keep trainers from battling others, as it would require a change in human nature.
Feeling sorry for herself and beginning to consider what she would need to do to resume Lance's plan, Yellow stumbled across Red, who was giving a kid a lecture in catching Pokémon. "Listen, Pokémon can be scary, but they're basically kind, loving creatures. If you take care of them with a gentle heart, then they'll always be your friends, understand?"
Yellow listened to her friend, her eyes wide in amazement. When Red had helped her catch Ratty, he had said the exact same thing. While Yellow believed it with all her heart, she had witnessed dozens of trainers who had trampled on that philosophy. But as the kid nodded, eyes shining with excitement, Yellow began to think. With a few heartfelt words, Red had convinced a child, who would one day become a trainer, to watch over his Pokémon with compassion and understanding. If that kid spread the philosophy to others…
Nowadays, Yellow tries to follow Red's example. She works hard at convincing others around her to love and respect their Pokémon, even though it hurts to know that putting her energy into this means spending less time using her powers to heal Pokémon. She keeps her eyes out for promising trainers and makes sure that they know to treat their Pokémon well. It helps that she lives nearby a gym – trainers are always coming to challenge the leader, and if they fail, Yellow is there for them with a smile, patching up their Pokémon and giving them tips on working with their Pokémon. She is also cautious about who she takes care of – if a look at a Pokémon's memories show that their trainer doesn't treat them well and the trainer doesn't improve – well, Yellow's skills with the fishing rod have not deteriorated, and sometimes a trainer will find themselves missing Pokémon around Viridian.
Yellow's goal is still impossible – she knows that no matter what, Pokémon will have to get hurt while they battle others. But she knows that if trainers treat their Pokémon well and try to be their friends, the hurt that Pokémon will suffer during battle will become inconsequential, something they are willing to suffer through in return for spending quality time with their friends. Yellow has learned that Pokémon can be hurt and still be happy through all her friends, both Pokémon and human, and even though she will never hurt a Pokémon if she can avoid it, Yellow understands that compassionate trainers will do their best to minimize the hurt.
Secretly, though, Yellow now has another goal she lives for – to assist her friends in their endeavors as well as possible. If Red needs something done in order to save the world, or Blue needs her to go out and be the cute face gathering information, Yellow is there. When Green asks her to watch over the Pokémon at the gym in his absence, or Chris is around, looking for a new strain of Pokémon, Yellow is happy to help.
Yellow knows that the goal of helping others as well as possible is not truly a goal, per se, but she is willing to give it her all. Because, in the end, even though she has had thoughts about wiping out humans so Pokémon didn't have to suffer, Yellow is in essence a very different person than Lance was. As any of her friends would say, Yellow is more pure, less fractured than Lance was, and she can see an alternative to eliminating part of the world so the other part will thrive.
And even though Yellow would like to be as strong as Red, or as focused as Green, or as intelligent as Blue, or (and she only admits this one to herself in private) as determined as Lance, Yellow knows that her gift with Pokémon was given to her for more of a reason than simply being born in Viridian Forest. Yellow is the purity of the group, the Healer, the one who sacrifices her own wellbeing to ensure that of others. No matter how many voices whisper to her that it would be easier, or simpler, or even more rewarding to take Lance's way out of things, Yellow will work as hard as possible to take the hard way, the complicated way, the less rewarding way to ensure that everyone can fulfill their happy dreams.
Yellow finds it depressing that she doesn't have a clear goal in mind. Yellow finds it to be unbelievably tragic that she doesn't have something simple like Red, or Green, or Blue, or Chris, or Silver do to work towards, and that she is stuck with the ambiguous goal to support, to heal.
And during those times, she becomes happy again quickly, knowing she wouldn't have it any other way.