Agree with the others that Club Med ski holidays are not cheesy, but OTOH if you pick the right (smaller family-run) hotel, it's often not much more work - once you're picked your hotel, they will likely refer you onto a ski school & ski rental place. Also: there are many ski hotels that offer up a comparably all-inclusive package, including childcare. If you are struggling to find a travel agent who is really helpful with European ski holidays, you could always use a UK ski tour operator like Iglu.
The biggest plus of arranging your own holiday is that if airfares are the biggest component, you can start your planning based on where you find the best deal for flights. If you book with Club Med, you will then be constrained by having to find a flight to a specific airport.
Also, going independent allows you to cater to the specific needs of your family - do you have fussy eaters or need more room to spread out, so that it makes more sense to rent an apt? Will your kids be okay joining in with tons of group activities? (Keep in mind that many hotels have attached holiday apartments or even have studios in the hotel itself so it's not even an either-or option.)
Flying to Europe for a week may actually work out cheaper than flying to any of the big name destination resorts in North America, not least because hotels are better value, lift passes are cheaper, and ski school is also cheaper. If you have either an Epic or Ikon pass, there's quite a few places that are included btw - although they tend to be the big-name expensive resorts.
As you may have seen from the news this year, Europe had an even worse start to the ski season than the East Coast resorts (temps in the 60s etc, no snow) although things reverted to ~normal during Feb & March. You'll need to keep that in mind in terms of resort choice - which places will still be attractive/fun places to be even if the snow isn't great.