The Insanity that is Paris Traffic

(move to around 6:30 in the video for the pertinent clip)
So I drove in Paris on my recent trip to Europe, and it's some of the craziest stuff I've ever seen.  What's crazier than anything else though is the traffic circle around the Arc de Triomphe.  Let me give you a little bit of a feel for it.  We came into the city on a road with 4 lanes of traffic going each direction.  Once you pass the Arc, it turns into the Champs Elysses.  This road intersects with 3 other major roads of at least 2 lanes each direction at the Arc, creating an absolute mess.  There are no "lanes" inside the circle, it's just one continuous stretch of asphalt.  As a result of this brilliant city planning, I had driven into a situation the likes of which I may never see again.  It was like the wild west of vehicular travel.  I tried to go around to my street in an orderly fashion, going around the outside of the circle.  While I was doing so, I had to dodge cars that would beeline from their respective roads into somewhere around the center of the circle, make a tight turn around the Arc, then beeline right back out.  There weren't rights of way, there weren't any rules, it was just every man for themselves in a massive free-for-all.  My wife was laughing at the insanity the whole time as I just tried to avoid getting hit.  We made it out alive, and I think there has to be a better way to do it.
For example, in many intersections in Paris there's a through traffic tunnel and local traffic split-offs.  Why not at least put the Champs Elysses right under the Arc?  You can route some traffic onto the circle, but a good part of that traffic is going straight.  Punch it into a tunnel that doesn't have to go through the circle.  You could even make it a standard intersection with the streets that are perpendicular to it.  Have a standard intersection below it and you could probably eliminate 30-40% of the traffic that is overloading the Arc's traffic circle.  Then it might not be such a madhouse.