When I was a kid (in the 1970's) about 90% of the horses I rode were kid safe, beginner safe. That's what everybody had. These were often fairly coldblooded animals. Now that did not mean that they did not have bad habits but none involved bucking or rearing or biting. They were calm steady animals to begin with or became that way. They were ridden a lot, but only by one or two people. Most of their miles were trail miles, which horses generally find a natural and relaxing way to move, unlike being endlessly ridden in circles by different inexperienced persons. Frankly I would get pretty sour myself in that situation.
I'm afraid you have a bad situation of poorly trained and managed riding school horses and no other options. If you want a patient steady well-trained horse mainly you have to borrow somebody's (who knows how to keep their horse that way), or buy one.
I used to ride school and rental horses on occasion and was invariably appalled at how dead-feeling they were. And these were the safe ones!
Agree with this
many riding stables have all kinds of riders riding those horses, plus those horses become arena sour, as how could they not, mostly, if not soleya rena ridden, giving lessons that are repetitive, over and over again, often on the same day
Even under the watchful eyes of a trainer, these horses are ridden by inconsistent cues, between riders of various ability level, and add that to where they are ridden, you have a very bad combo, far as producing, or maintaining an honest beginner horse, or honest horse-period!
The horses I felt confident letting my children learn to ride on, were horses we raised ourselves, that I trained and rode myself for several years, and that I still rode, after my kids were riding them
If you can find a horse that has been ridden lots, by some youth, who has just out grown that horse, or a ranch horse that has been a working horse, ect, buy that horse, then take lessons on the horse, you will have better luck in finding that horse that will help you develop confidence, versus one ridden my many different riding students, doing the same arena exercises over and over again.
Even the best show horse, ridden by an experienced rider, will become sour, if never ridden out