I'm not sure if I would call that cat like reflexes, when you're riding in those type of conditions you were expecting and are prepared to dismount the bike quickly, had that happened lets say on black ice that you were NOT expecting you would have more than likely crashed and burned.
Also riding on ice can be helped by using studded tires, I know you were doing this for fun but CORZHENS was questioning the sanity of it all, but in reality with studded tires it can be done real safely. You can read about the different types and uses of these tires here and select the set that would be best for your situation:
http://www.peterwhitecycles.com/studdedtires.php
Of course keep in mind that studded tires should never be used for the long haul on dry pavement, if you're switching from snow to ice to dry and back again in one ride then it's ok, but to go out on a dry condition day would wear the studs out too fast so you have to change back to your regular tires before going out. I'm not sure what the tire pressure markings are on those studded tires but typically you would want to use about 10 to 20 psi less than normal due to cold making tires real hard and stiff which decreases traction, it's possible those tires have already taken that into consideration and the recommendation is already on the sidewall, and or the rubber they use is real soft so when cold they perform correctly.
I use to ride on ice when I was a kid for the thrill of it on a 65 Schwinn 3 speed Racer, I can't recall the tire size exactly I think they were about 1 3/8ths inches wide seems to ring a bell, but they weren't as wide as a mountain bike tire, and no studs of course. I don't even recall falling down, maybe, but I don't think I did, but I remember sliding a bit trying to gently turn but I was prepared for that anyways so I caught myself without ever hitting the ground! I ran off the road a bunch of times but would simply end up on snowy grass surfaces which I had to dismount because the tires were too narrow to track on snow, of course I wasn't dealing with ruts, the ice had fallen overnight onto the pavement, they closed school because of it so out I went, I had more trouble walking then riding! LOL!! Of course I knew to use only the rear brake which was also fun sliding a long ways. I did learn how to release the brakes to regain control and not to make any sudden change of direction from those experiences. I'm 64 years old now so I don't take unnecessary risk like that anymore though I do ride on fresh fallen snow with no ice with just standard MTB tires without issues. I won't ride far on snow only because I don't like to be out in the cold for too long, so about 20 minutes is good enough for me, and since I don't go out to ride a long ways in those conditions I don't have studded tires.
