well, last question first, never make your plot "more like" anything. When you have the freedom to come up with whatever you want, making it like something else you already know or is popular seems like a waste, you know?
I like the story line. But how would you start off? it seems a little odd, story wise, to pass months of training to be able to use any pokemon, and for your first mission you get a level 5 starter. Maybe, add the training as part of the game instead of something inferred. Hell, you could have the majority of the game in training. Make the gym leaders into your teachers/masters, who you have to defeat to move on to the next lesson/training phase or something. And then have the missions be extremely tough post 8 leader dungeons. Or not. It's your thing. But if you do start after training, then maybe you should give some reason or plot point as to why your pokemon is so weak and there's only 1. like it's a tradition or something that they receive one newborn partner after training, but have to raise them on their own to prove themselves.
These are just my ideas, do what you want. If you don't mind though, and don't do it yourself, I might use something like this myself though... I like the idea of a school like environment for training elite pokemon trainers.
I like your idea of thw game being the training session. I wasn't planning on having the Pokemon weak. It is war. They were going to be atleast level 20. But I do like your idea still.