It would be my opinion that this horse is not 'broke enough' to be going over fences. This horse is not 'between the rider's reins' and is definitely not 'between the rider's legs'. This is part of teaching a horse to guide properly. For me, it is a prerequisite for jumping even small jumps at the trot much less going over bigger fences at the canter / gallop.
You must be able to effectively guide a horse through small and large circles. These circles should be round, not egg shaped or with square corners. Then, you practice cantering 'squares'. I teach squares, out in a large pasture, with the horse on the inside lead making tight, well-ridden rounded corners with the horse's head slightly to the inside on the corners and its head straight in front of it on the straight sides. Then I teach cantering squares on the outside lead with a small, round 3/4 circle to the outside at each corner. Again, I teach straightness on the long sides and a small, round circle at each corner.
When I teach a horse and a rider to lope round circles and straight sided squares, I use cones or markers so the rider can aim for the markers. I like teaching the rider to ride through 'gates' made of 2 cones that start out 8 feet apart and end up 2-3 feet apart. The rider should be able to guide the horse through these 'gates' with the horse's head as straight in front of it as possible, or with a slight bend to the inside, if the gates are located on a circle. This tests the rider more than the horse. If a rider is not able to successfully guide a horse through these narrow gates, the horse is not broke enough -- is not responsive enough to the rider's legs and rein aids to stay centered. When the horse can do this while being rated from a slow canter to a 2 point gallop and back to a slow collected canter, it is NOT going to run out on fences, miss turns or flags and will be well rated and under complete control.
Going over fences with a horse that is not broke enough is like running barrels or roping off of a horse that is not broke enough. It is a wreck looking for a place to happen. It is riding without the control and the foundation to keep things headed the right direction - literally. The rider does not have the tools to fix a small problems before it becomes a huge one.