I understand they make the child higher which makes the seatbelt fit better, but apart from that it doesn’t really do much in terms of preventing injury to the child.
You say this, but this is absolutely huge and makes a big difference in preventing injury to the child. This is what I mean when I say parents don't generally understand what the job of a booster seat is. First of all, if you have proper lap belt position it will anchor the pelvis properly, which means that the person does not slide around, they stay where they are, which is the single most important job of a seatbelt. The main thing that you need to avoid is hitting your head on anything in the car, because (as you know, sadly) head injuries in a car accident tend to be very bad news.
Secondly, if the lap belt is too high, it tightens around the body around the tummy inread, so not only do you have the person restrained by a much more flexible body part so they are more likely to move further before stopping which is worse for injuries, the part of the body which takes the impact is the abdomen which is soft, squishy, full of important stuff and much more likely to get hurt than the pelvis. It puts strain on the spine and the internal organs. Children who have shorter legs than adults often do not sit straight up at the back of the vehicle seat because they would not be able to bend their knees and this is uncomfortable, so they slouch to fit correctly. is a bit exaggerated but shows the difference in restraint between a seatbelt and a booster cushion. Booster cushions have basically not changed in design since the 80s when they were introduced which, yes, it does mean they are primitive compared to a well designed modern HBB, but it also means that they do the job they were designed to do extremely well and you can't really improve on the original booster cushion design. This part of a booster seat is the same today as it was 50 years ago.
And then if your car has curtain airbags, bringing the child into the area of effect for these too. And improving comfort of the shoulder belt, which helps reduce practices like tucking the belt under a child's arm or behind their back, both of which are very bad but are quite commonly done when a child is taken out of a booster seat too early.
The side panels do not actually always do much and sometimes they are completely useless. A flimsy piece of plastic does not make much difference to an impact metal door. If your seat is designed well then it will have other layers such as polystyrene, EPS foam, memory foam, or blocks which act as a crumple zone or brace, but the sturdiness of the backrest is also important. The Britax campaign is a little misleading, because they are showing their own HBB which is designed extremely well and does a good job in supporting the upper body and keeping the shoulder belt in a better position. A lot of HBBs on the market do not do this very well. For example, this is Maxi Cosi Rodi, which was actually well rated at the time - this is a VERY old video, from 2002, but it's a good example and some cheaper HBBs are still designed like this. You can see the movement of the dummy is quite similar to the Britax booster cushion.