I'm going to have to disagree with you more than once.
1. Lillie was traumatized due to what happened with Sivally (Type: Null) and Nihelego. It's understandable that she would be afraid to be near Pokemon
***NOTE*** I'm probably very wrong, but wasn't after she was traumatized, she was wanting to be friendly with a Pokemon, she was in danger, and a Pokemon trying to warn her ended up terrifying her further, OR was that an entirely different character? It could've even been a boy. I can't remember, I remember someone who was apprehensive about Pokemon, wanting to give them a chance, then terrified of Pokemon because they though the Pokemon in question was trying to attack them, when actually it was trying to warn them and I think theyy ended up not liking Pokemon or being distrustful to them. Why can't I remember who that was?
2. I have to disagree with your idea that a traumatizing event in someone's past ONLY affecting them one specific way, there is at least one example (which we do not need to get into details) I can think of when something bad happens to someone at a very young age, it will likely change their entire personality and who they are a s person when they become adult, and not necessarily in context with what the bad thing that happens to them. A more simpler and easier to talk about (yet a very hypothetical) example, is a child's balloon being stolen and the event was so traumatizing (for hypothetical reasons) that the kid grows up smacking people across the face any time they get close to them. Doesn't necessarily have to do with balloons or possessions being stolen. But it was obviously a factor (yes I know it was ridiculous but I did say very hypothetical). What I'm saying is, if there is any truth at all to Disney's "Inside Out" each event that happens in a person's life shapes their future. In Lillie's case being absolutely terrified by Pokemon, would absolutely affect how she interacts with not just Pokemon but humans as well, like developing a fear of these two things potentially lashing out her would make her reserved. I also imagine repressing her memory of the event, would also cause unintended consequences of her being unable to properly be affected by her traumatizing event that affected her, what I'm saying is, she is affected, doesn't know why she's affected and as a result of not knowing why she acts like she does, could explain why she's timid, reserved.
I don't know about the whole "sheltered" description however. That sounds like an absent Lusamine, and an over protective brother thing more so than being traumatized by Pokemon. But timid? No, it does make sense her trauma of what happened would make her generally timid. It wouldn't just make her timid of JUST Pokemon.