random23, I know your question is directed at Bill, but thought I'd chime in as I considered anesthesiology myself.
The short answer is yes, there is a place for anesthesiologists in private practice. However, there are very few who work in that sort of environment. Places like the Animal Medical Center in NYC, Ruffian Equine Medical in NY, and Animal Scan (a private MRI company) employ anesthesiologists. I've also heard of a couple who have established pain clinics, and when I was considering the possibility myself I thought of having an independent consultation service. i.e., folks like veterinary dentists with difficult patients would provide me with patient records, PE findings, ECG etc and I could consult with them on protocols...not sure of the liability of that given that I wouldn't actually be seeing the patient. I also thought of an ambulatory anesthesia service--having scheduled days with various specialty practices who could operate on their high-risk patients on the days that I'd be there. Unfortunately, though, you can't always schedule a surgery for a high-risk patient--they're often emergent scenarios.
I spent some time working with one of the aforementioned private practice anesthesiologists and decided it wasn't for me because of limited/absent client contact, very limited to absent patient follow-up, and no relationship with the patient once it's extubated and out the door. For me, that relationship was a bit too brief to be wholly satisfying. However, I was interested in it because of my fascination with pathophysiology, pulmonary physiology and pharmacology--I still think it's a really cool specialty and I had a fabulous time with the anesthesiologist.
One of my specialist friends pointed out to me that as an anesthesiologist, all of your job opportunities are necessarily dependent on other services/practices/doctors--which is true in the case of any specialist, but more so for this particular specialty. Something to keep in mind.
Hope this helps. 🙂