First of all, you have to consider that most(certainly not all) baby trainers want the babies broke quickly, in the least amount of time possible. This doesn’t allow for lunging in a round pen, or line driving, or anything else even remotely time consuming. Generally, this type of breaking involves turning the each rider loose with a horse in a standard 12X12 stall, and letting him or her go at it. This allows for breaking six or more horses at a time, depending on how many riders are available. By the end of the morning, every baby in the barn will have had a rider up and turned in the stall.
The first day, the rider will usually sack the horse out, put on the bridle and the saddle, and hold the girth tight with one hand, not actually fastening it yet. Then he turns the horse in the stall, with a rope shank attached to the bit or to the halter. By not fastening the girth, if the horse comes unglued, the saddle can be removed quickly, instaed of sliding under the belly, as it might if just loosely fastened. Once the horse accepts the girth, then it is tightened by degrees, turning the horse both ways at every step. Then the rider will jump up and “belly” the horse, until he accepts the weight, then he will get astride and sit. Next comes turning with a rider. Some trainers will have a groom turn the horse on a shank with the rider up at first, some let the rider do it themselves, using the reins, their stick and their heels. Most riders I know would rather tough it out themselves rather than risk their neck to a groom who might flip the horse if it acts up. This often all happens on the very first day, with two or three more days of turning in the stall, and then straight out to the shedrow and/or paddock. If a horse is particularly bad or nervous, the rider may take longer, but the average horse gets that far in a single session at most training centers.
The idea of having the horse in a stall for all of this is that not only is a horse usually more comfortable in his own stall, instead of out in the open in a strange place, but a round pen gives them way too much room to get away from you, and it gives them way more room to buck. Trying to do all that in an open round pen all at once would be far more dangerous than in a stall, as long as the rider knows what he’s doing. I personally cannot imagine climbing on a brand new baby the first day in an open round pen. Give me a nice enclosed stall any day of the week. We used to break around 130 babies every season, and in 7 years I only remember two riders getting hurt. One was when the groom did manage to flip the baby over on top of him by yanking down on the shank while he was up, and the other was a rider who refused to put a shank on the horse at all, and it got away from him when he put the saddle on the first time, breaking hs finger.
It’s all what you’re used to. You just have to be able to adapt to the situation.