It's a bit of a grey area for me, certainly not as straight-forward as wear a body protector and you won't be hurt in a fall (or don't wear one and you will), you need to do your own risk assessment really and it's a tricky one because the absolute risk (how likely you are to be seriously injured in a riding fall) is very low, there are very few fatal or very serious injuries in riders every year, riding is overall a safe activity, but the relative risk (how likely you are to be seriously injured riding compared to sitting at home watching TV) is high, on this metric riding is very risky indeed - then you need to look at how likely a body protector is to reduce this risk and here it gets more complex as you also need to define for yourself what a serious injury actually is, body protectors do little to protect the most common kinds of injury e.g. bruised, sprained or broken limbs, concussions but do quite a lot for some of the more serious types of injuries that affect the torso and major organs. But you do then need to consider other factors too such as type of activity (fast riding in open outdoor spaces and XC the most dangerous, going around a school in walk and trot on a bombproof horse the least, again consider relative risk though, the vast majority of XC riders whether wearing a body protector or not don't get injured and there have been very rare cases of broken necks coming off in an indoor school at a low pace). The best study I've seen looked at cross-country riding only (so obviously at the more dangerous end of the spectrum) and suggested relative risk of injury was reduced by wearing a body protector by 53%. Other studies with different designs suggested a reduction of somewhere between 20-80% so a huge range. I do have to say if they were at the upper range of that effectiveness you'd think they would have caught on much more in other similarly risky sports - BMX, skiing, mountain biking, motorbikes etc where you see similar kinds of accidents to riding and where people do already wear some protective gear like helmets, but they really haven't at all so far as I know - which brings me on to say....
You do have to consider the negative impacts of wearing something constricting and uncomfortable - I've had this debate with friends and some people swear blind a body protector isn't uncomfortable to ride in but frankly I think they're mistaken and/or have gotten so used to wearing one all the time they forget how much nicer it is without one - design and comfort has improved a lot over the years and some of the more high end models are definitely much more light-weight and breathable than they ever used to be but in the end part of the way they work to protect you is by creating a semi-rigid structure that holds your organs in place in the event of a fall and so they have to be tight fitting and somewhat restrictive. I'm not saying that makes them not worth it, I think similar comments could apply to wearing a helmet and largely people agree you should do that, or to take it outside riding, seatbelts in a car are again a bit uncomfortable but nearly everyone agrees the benefits are worth it. But you have to take it into account at the end of the day.
Personally when my kids were riding I set the boundary at body protectors needed for jumping, hacking or XC - anything more than schooling basically. I have to admit to being a little bit more lax with myself and I don't always wear one for low level jumping or a quiet hack in the summer if I judge the risk to be low. Like I others on the yard do wear one all the time no matter what they are doing which is fair enough too, and a few people never do. So long as you've thought about it and come to your own decision and you are abiding by yard/school rules then I think you are good!