Don't give up on that goal; those horses are definitely out there and they are not necessarily $$$. My first horse, a grade horse, mostly QH, was $1K. He was a true "all-rounder" and would hack out, jump, and competed through 1st level. He also could do tempe changes and canter pirouettes. My Trakehener ($6K) could hack on the trails with a rank beginner (my husband), was trained through second level and hunted in a bitless bridle. Yes, that's out of your budget but for the breed was quite reasonable. My OTTB ($300) hacks on the buckle, foxhunts first flight and easily has the movement to go beyond second (I don't show any more, but I do school him in dressage).
It seems like a horse that is calm enough for me, can be ridden on trails nicely, but also turn around and show with does not exist. The horse is either good for show or good for trails.