John, was it your car or a rental?
People can, and do, get just about everything back there, but I wouldn't (not couldn't - wouldn't) do Bighorn in most modern passenger cars, especially if they belonged to me rather than the rental company. Anything with a bit of ground clearance and a competent driver, however, will be fine, barring washouts, bad weather or inattention. 4WD is definitely not required under normal circumstances but as always with backcountry roads, after a wet spell all bets are off as there are places where mud-holes can form and the road can severely rut.
Living out here and enjoying the backcountry, I've always owned some sort of reasonably capable 4WD, not so much for the more extreme stuff, though I enjoy that too, but so that I can go to places like Bighorn and not worry at all about what I'm doing to my vehicle. For anyone flying out to visit the C&TS and wanting to do a serious chase on the east side, I'd recommend some sort of modest SUV as a rental. It doesn't need to be anything extreme, just something comfortable in the dirt.
Again, it isn't a case of can't, but speaking of can't - in my years crawling through the backcountry, I can't tell you how many times I've strapped onto "that guy". The guy who loved telling his buddies all about how he can get his vehicle just about anywhere. Right up until it bit him in the undercarriage.
Scott