I am privilege to have seen the Spanish Riding School perform in London on three occasions. The horses aren't very tall but are strong with compact backs and good bone.
They do not have the 'big' movement that the modern dressage horse does.
As for temperament I cannot say but, doing the short rein work, (trainer walking with only one rein attached to the horse to perform unmounted airs above the ground) one horse was getting very agitated and was jigging and uptight.
The Head could see what was going on. The horse was meant to be doing a Capriole but all it did was charge forward.
Very quietly the Head swapped horses, he got him softer and more settled, when asked to do a Capriole the horse did a slight jump forward in the centre of the arena. The Head pulled him up and took a treat from the top of his jacket at the back, gave it amd a pat amd went on. After about four tries which were nothing spectacular, the horse leapt into the air above Head height and his back legs parallel to the ground. Again he was stopped and praised and walked to the exit and was taken outside.
I did do the stud work and riding of an Arab Lipizzaner stallion cross, he was good temperament but nothing special to ride.