I think you should let players use training points to lvl them up to the next stage
I don't think many would use that option. Training points are used for spells, and even if you save them, you will need them later on in the game. Special training points for pets might work, but I don't see KI changing the whole system.
I agree with you, but Ravenlady made a very good point there. Training points are made to be trained in specific schools and spells, not to speed up the process of training a pet. Maybe what you're trying to say is that we should have special training points that are specifically made for the purpose of training your pets.
The problem is how would we get the pet training points? I don't think Kingsisle would change their whole scheme from the very first world just to add something little like that, even though it would be pretty special. You have a very good idea, but I reckon Kingsisle should put in a potion (or whatever they're called) into the Crowns Shop that can be purchased with crowns as well as gold. This can do exactly what you want it to do, it will also save us time, and for those who like to spend their money in the game, it will also get Kingsisle some more money.
If you moderator is reading this right now, please tell the staff of Kingsisle to take this onboard! It would be a great benefit to have, and we also wouldn't have to waste so much time and Pet Energy! Great idea, ~Maddie