CAAs are trained by physicians and CAAs, typically in a medical school environment. Every program has a medical school sponsoring institution, and many of the programs have very strong ties to medical school anesthesia departments. Some do in fact take medical school classes, others take some of their programs with PA students, while others have totally separate courses - it's entirely program dependent.
This "trained to be independent" argument from CRNAs is a crock. CAAs are, by training and law, a "dependent" provider, meaning we have a formal legal relationship with an anesthesiologist. That being said, EVERY CAA can, by training and law, act without the presence of an anesthesiologist in an emergency situation. My very first on-call case was a twin gestation ruptured uterus. I was the first to arrive, even before the OR staff, with the OB doc screaming and banging on the locked OR doors. I had pre-oxygenated the patient and was pushing pentothal and sux before the anesthesiologist got to the OR. Every CAA has the ability to assess a patient and make an anesthesia plan - of course in a care team environment, the physician is and should be the boss.
CAAs should be able to practice in every state. Of course the main reason we can't is CRNA opposition, either individually, or through their "professional" organizations. The other is that student nurse anesthetists are still widely used as free labor in anesthesia departments across the US. A lot of practices simply won't let go of that concept.
Happy to say that CAA numbers are rapidly increasing (nowhere near CRNA numbers, but far better than 25 a year combined with the original two programs). There are a number of programs in the development stage, and several who are accepting applications for their inaugural classes in 2022. The limiting factors are simple economics (starting a program is not inexpensive) and logistical (a program has to have adequate clinical sites for it's students). You want more CAAs? Offer to become a clinical site. CAA students can rotate just about anywhere - medical education laws in most states have provisions for all sorts of clinical student education, both for physicians and non physicians.