IMV Chamonix is the better bet for the better skier. I'm not so enamored of the town, but there's nothing wrong with it. Snow cover should be good. If you're looking to go back to the States having said you skied something World Class, look no further (Verbier would have similar cred, though IMV the town blows). I don't want to come down on Cham, but it's hard to compare it to Zermatt in terms of convenience. Once you get to Zermatt, you'll never be in a car, you won't need to transfer between different lift stations (maybe just from one side of town to the other). Cham definitely has the extreme skier vibe going like nowhere else I've ever been, except maybe Verbier and Jackson Hole.
Zermatt is a fabulous place to hang and very easy for groups who need to meet up at different times, all pedestrian, lots of bars, very atmospheric. Of Europe's large ski villages, I think it's the most attractive. The scenary is unbeatable and snow cover probably slightly more reliable than Cham. There's a lot of good expert skiing and offpiste, but again, not as much as Cham. Also, Zermatt has the link to Cervinia, which is even more snow-sure and has miles and miles of great cruisers for the intermediates in your group. Italian food too.
I echo the earlier post on St Anton (adding in Lech and Zurs, which are in their way even more charming and have underrated skiing and even better snow). The Arlberg zone is Europe's champ in terms of snow cover (OK, maybe stripping out ultra-high destinations like Tignes or glaciers). I keep going back to St Anton and love it more and more every time. It's just a captivating place, ski-wise and in terms of feeling. While nightlife can be good in Swiss or French ski towns, that's mostly due to foreigners. The only country in the Alps where the locals really know how to have fun (my view) is Austria. To me, the Arlberg is just a winning combo: great snow, lots of expert terrain, great atmosphere, good food, decent value, excellent ambiance, fun vibe.
All three of these resorts are really easy to get to: Cham from Geneva, St Anton from Zurich, Zermatt from either (or from Milan). Cham probably easiest of the three. Zermatt is definitely the most expensive. I think St Anton's the most snow-sure (but the season's not as long as Zermatt's). Zermatt has the easiest logistics once you're there. Zermatt probably also has the fewest Americans and Brits of the three, in case that's a factor.
I hope this was helpful.