Bareback horse riding lessons are not illegal!
Individual stables may not offer them, or may offer them only after the student is proficient with a saddled horse, but there's no law against it.
The question is, why don't they offer it?
The student is more likely to be injured bareback than with a saddle. Law suits, as you say, are part of this.
Therefore, stable insurance policies may not cover it, and thus the stable cannot offer it.
Therefore, stables may feel it is too great a risk a risk to the student and/or to the stable (liability for injury).
But it's not illegal at all.