I think what Corporal is expressing is what a lot of us are thinking. You don't think you are a beginner, but you really, really are. It takes years of riding to get past the beginner stage, not months. My daughter has been riding for almost 5 years and is just now considered a novice (she's jumping and has mastered all the gaits). You say you need more confidence in the jog, but if you're not a beginner, jogging would be just as easy as walking to you - you wouldn't even think twice about it.
We also have an Arab. I learned to ride on an Arab when I was 11. Neither was ever stubborn, just very intelligent. If the horse isn't doing what you want, then chances are, you aren't doing a good job communicating with it. Arabs are incredibly intelligent, sensitive horses.
You may not like the term "babysitter horse". Think of it more like an experienced horse that you can learn on and make mistakes on. Because you will make mistakes. A lot of them. And making a mistake on a green horse can lead to disaster. Whereas making a mistake on a well-trained, experienced horse can be forgiven. They are patient that way.
We aren't trying to diminish your skills as a rider or horse person, just trying to prevent you from getting into a situation that will be bad. You seem to be assuming a lot of things (that your trainer will board your horse, that her students will train it) which are red flags for us.
When I was 12 I got my first horse. I'd had ponies since the age of 5, but they were all boarded. My first horse was a Quarter horse who was a good horse in a lot of ways, but had been in an abuse situation in his previous life. My parents were not horse people and although my coach came with us to look at him, in retrospect, he was not at all the horse that I needed. He had a lot of strange fears. He was afraid of water, anything that made a hissing noise (even fly spray, so I had to sponge him with it instead), the hose, puddles even. He was terrified of cars, which led to many dangerous situations when I could have been seriously hurt (the only thing that saved me was that I could really stay on a horse back then). He did not like trails because at every corner, something spooked him (deer sent him reeling back towards the barn at full gallop once, like they were lions or something). In short, my idea of having a horse of my own in my backyard that I could love and dote on and spend countless hours riding on trails with was severely dampened. I do have great memories of him, but I remember that time with him as pretty stressful. It took me 20 years to have horses again.
We are just trying to save you from the unpleasantness that can come with a horse that is unpredictable and potentially dangerous. A calm, quiet horse you can keep working with as you progress as a rider (and trust me, it takes many, many years) is what you should get. Pay a little more now and save yourself some money and heartache later is my advice.
But as Corporal says, if you choose to get a rescued horse or a green horse, then that is your decision.