What you are asking is eminently doable. You just need to know the secrets and how to find them.*
Assuming that you want to stay in an apartment rather than a hotel, and that you are happy to organise travel yourselves, then I would recommend Austria or Italy rather than France or Switzerland. Pretty villages that are not purpose-built resorts but actual working, breathing places where people actually live have a much better atmosphere than the French mega-resorts. Prices are much more reasonable than in France or Switzerland. But - don't be tempted by cheap places such as Bulgaria. There are reasons why they are cheap (and the reasons are not good).
Have a look at this old thread: >www.mumsnet.com/Talk/skiing/3893018-First-time-skiing-how-does-it-work
particularly the post from @TeetotalKoala about 9 posts down.
I know Austria really well, so can suggest somewhere like Saalbach-Hinterglemm. Ski schools are excellent. There are walks, snow-shoe hikes and sleigh rides for non-skiers. Zell am See is a short bus-ride away for a day out in a bigger town (but Saalbach is far better for skiing). Every cabin lift has a restaurant at the top - non-skiers can buy a special cut-price pass to use the lifts so you can meet the kids for lunch or just find yourselves a hut or terrace for a long lunch. Plenty of accommodation in the villages - apartments/B&B/hotels - in all price ranges. Easy 90 min. transfer from Salzburg airport, which has numerous flights in winter from the UK (including Manchester) and Ireland. High enough to be guaranteed snow (the beginner pistes are up on the mountain rather than down in the valley) but not a glacier. Speaking of which, what does anyone want to do on a glacier in late December when the temperatures can be minus 20 and there is nothing for non-skiers to do except stand and freeze? Don't think of Salzburg as a ski destination - it's miles from any resorts and is only really the entry airport. Get your accommodation in an actual resort such as Saalbach, so you've only got 5-10 minutes walk to the lifts.
Going to Chill Factore is an excellent idea. There used to be a "Ski in a day" course which is well worth doing to check that the kids actually like it. Alternatively, a few shorter lessons will put them ahead of everyone else once they arrive in resort.
DIY is really easy if you know the ropes. If you don't want to spend £1800 on the latest designer ski suit (you don't), there are perfectly good ski clothes at Decathlon or Trespass - or borrow some from the secret club members at work! Buy proper ski socks and ski gloves, and ski thermals. Decathlon or Aldi are your budget friends. Everything else you organise in resort - skis, boots, helmets, ski school, lift passes etc. Your hotel/host will know exactly where to send you for all of these and you won't need to pre-book at Christmas.
*P.S The secret club meets over at Snowheads.com