A large portion of my family are doctors as well, so this is anecdotal from them and also from my own experience having rotated with them and with residents. Medical students may touch the patient, but they don't make decisions. They shadow, hold retractors, maybe make an easy cut, do a few stitches, etc. I am sure your daughters have been present when a baby was delivered---maybe they even caught the baby. I doubt they were leading the birth process, and I doubt the attending was not hovering over their shoulder. They do parts of procedures under lots of supervision. The residents are the ones that are supposed to be learning, and so in the clinical settings, medical student experience often comes second.
As for anesthesia, first year interns come into anesthesia with very little anesthesia training (maybe a month or so with certain med schools). They don't know really how to prime IV tubing. They don't know any of the anesthesia meds other than a few lectures on the anesthetic gases. Yes, they have extensive medical training. Do you need to know tissue pathology and genetics to do anesthesia, though? No. A large chunk of their training, what they're tested on in Step 1-3 exams, is not necessary to their practice in anesthesia, whereas CRNA school is focused only on anesthesia. Do I think at the end of anesthesia residency, doctors often have more experience and medical knowledge? Yes. But largely because most residents I know work many more hours than the SRNAs. Even still, do I think that's training that is necessary to be an independent practitioner? No.
You can sit here and compare an ICU nurse to a medical student to a resident to an SRNA all day. At the end of the day, CRNAs do practice independently---completely independently---in many parts of the country. I live in Baltimore and there are facilities and hospitals here that are independent practice/bill QZ. Are people dying left and right because there's no MD? No. Anesthesia is very safe in the US and you put yourself at more risk getting into your car and driving than you do going under GA. At the end of the day, practice experience is what counts. (The art of emergence, for instance, is something that you develop on your own. Attendings even tell residents that it's not something they will really know how to do until they're out of residency and practicing on their own with less oversight.) Personally, do I want an AA, a CRNA, or an MD doing my anesthesia? I want the practitioner with more experience. I would want the AA with 5 years practicing over a CRNA with 1. I would want a practicing CRNA over an anesthesia resident with an MD behind his/her name.