I use it for trail rides (just walking and being relaxed) where I don't really need the control of a bit, and once or twice when I was giving pony rides to a friend's two kids and didn't want them to be able to pull on my horse's mouth. I've not used it for working in the arena.
My sense is that it with the majority of horses it doesn't give as much control as a standard bridle with a bit. (I say "the majority" because I know there are top-level competitors who ride and compete bitless and have just as much control as they would with a bit. But I don't think that's the norm.) However, if you make sure that your horse knows that just because she's not wearing a bit doesn't mean that she gets to ignore you and misbehave, I think it might be a nice workable alternative.
Also, you will need to give you and your horse some time to get used to the different setup. With the sidepull, you get pressure on the bridge of the nose when you pull back with both hands. (It is therefore important to make sure that the sidepull is adjusted correctly, so you are putting the pressure on the bone and not the soft tissue part of your mare's nose.) It's different than how it feels to the horse when you pull back with both hands using a bit. At first she might think "what's this pressure on my nose?!" but I think if she hates (fears, even) bits as much as you describe, she will quickly learn to work with the nose pressure instead.
Here's a good overview article about sidepulls: Side Pull Bridles - What is a Side Pull
ETA: If you want to try the sidepull concept, you can do it on the super-cheap by hooking lead ropes to either side of a good-fitting halter.