That's not quite true - I do ride on the rail, but only sometimes. Here's why - there are a couple of reasons. I think a lot of the problems we have with our horses are due to momentary lapses in our own attention to the horse, and lapses in our providing direction to the horse. We need to be there for our horses - how can they have a continuous conversation with us if we're not there? If I'm riding away from the rail, it helps me stay focused and attentive - I can't just mindlessly ride around the arena on the rail, I have to give the horse direction. This also means that we're doing things together - figures or riding to a specific point - which gives the horse a "mission" - horses love having jobs they can focus on and do well together with their rider.
And here's another reason I like riding away from the rail - horses are shaped like this:
Note that the horse is narrower in the shoulders than in the hindquarters. The horse in the picture isn't travelling straight - the rail side of its body is parallel to the rail but due to the horse's shape this means the hindquarters are travelling slightly to the inside, and there's also likely to be a slight bend of the head and neck to the inside. (Aside: are there other horse people out there who, like me, love those pictures of dressage movements in books where little diagram horses move around the figures?) If your horse travels like the one in the picture, rhythm and impulsion will both be problems as the horse isn't straight. Watch people riding their horses on the rail - I don't care whether English or Western or in what discipline - and you'll see a lot of crooked horses - it takes a lot of attention to ride a horse straight when travelling down the rail and most horses end up like the one in the diagram.
And when I'm riding away from the rail, I can't use the rail as a "crutch" - the horse and I have to travel with intention and if we're going to be straight, it's because we intend to be straight. If crookedness and wiggliness are an issue for you and your horse (and it's never just the horse), then riding in straight lines away from the rail, with impulsion and a specific destination, will in my experience do a lot to make things better. And straightness isn't a matter of steering - it comes from the hind end. A horse that's braced on the front end - either due to the horse or rider bracing or both, or ridden in a way that constricts the front end like rollkur, cannot effectively use its hind end to carry itself and cannot have proper impulsion - and its proper impulsion that leads directly to straightness and rhythm. Proper impulsion also cannot exist without softness and suppleness, and straightness also comes from the development of this softness and suppleness through all softening work including the use of figures such as circles and serpentines.
I also don't use the rail to teach the beginnings of lateral work, such as side pass. If you teach your horse to do side pass facing the rail as a barrier - that's what you've done - taught your horse to do sidepass if the rail is there. I find it's better to allow the horse the freedom to move - and to make mistakes - that being off the rail provides, and the horse learns the general principle rather than a specific case.
Now I certainly understand that, if you ride in an arena when lots of other people are riding, you may not have a choice about riding on the rail. But even in circumstances like that, it may be possible to do some things to engage your mind and that of your horse, and to work on straightness, like riding the quarter line, doing diagonals or partial diagonals, or leg yielding away from the rail for a few steps, riding straight for a few steps and then leg yielding back to the rail. Be creative - there are all sorts of things you can do. Cones are very useful as focal points when working off the rail. And most importantly, have fun!