You are right, it is pretty straight forward. Here's is what I get from it: CRNAs provide anesthetics to patients in cooperation with surgeons, anesthesiologists, dentists, podiatrists, and other qualified healthcare professionals. In cooperation with means that CRNAs are operating under the anesthesiologist's direction just as AAs. It's just sugar-coated to say "In cooperation with" instead of "Direct supervision of anesthesiologists". Now, if an anesthesiologist were not available then you guys pretty much work independently in delivering anesthestics.
Comparing AAs and CRNAs by the ability to work independently is only done to bolster the CRNA's title. If I was a patient and required anesthesia this is not the distinction that I would rely on. It would not matter one bit to me who can practice independently or not because all I would require from them is quality care. And to be honest with you, the one who is being supervised by the anesthesiologist or have the anesthesiologist breathing down their back as some CRNAs would phrase it, just might be the one to deliver the care I deserve b/c it could mean less risk of injury to me by having two anesthetists instead of one.
Anesthesiologists may be present during certain times that an AA is delivering anesthesia. Anesthesiologists may be present during certain times that a CRNA is delivering anesthesia as well. It depends on your level of experience. If a CRNA is fresh out of school they are gonna be monitored closely just as well as an AA until a certain level of trust has been established.
So, the difference that I have come to understand that some of you CRNAs boast about is being able to work independently (in areas where no anesthesiologist is present), and the fact that you guys are recognized and accepted in all 50 states. Congratulations! One day, hopefully, you will be congratulating AAs on their recognition and acceptance and not fighting against them.