If you can do some of the legwork yourself, this means you're not paying the tour operator for their efforts and are thus saving on their mark-up (or the hotel/ski hire/ski school might give you a discount, especially for cash, rather than paying commission to the TO).
There are hundreds of resorts and properties that are not featured in the TO brochures, so finding a good one is easy enough if you use forums such as this one or Snowheads where you can ask relevant questions.
If you've been using Mark Warner, then you will probably have been going to the expensive, well-known resorts, so there will almost certainly be less expensive options. [Not a criticism of MW, who are by most accounts very good, but if you no longer need their particular services such as childcare, then other options might make more sense].
You mention full-day ski school. This is the norm in Austria and Italy, where the kids tend to ski from 10am-12 and then 13-15. They can always miss a day or an afternoon if it is too much or if you want to ski a bit with them.
TeetotalKoala has an excellent post on this thread describing a typical Austrian holiday: www.mumsnet.com/Talk/skiing/3893018-First-time-skiing-how-does-it-work
Self-catering is almost certainly a cheaper option than hotels. British catered chalets are going to be very hard to find/very expensive next season as Brexit has made that business model unviable, if not impossible.
Some places to look at: La Rosier (FR) Kronplatz (IT) or Saalbach-Hinterglemm/Leogang (AT), Zell am See (AT) Maria Alm (AT), Ellmau (AT). Start with the local tourist office websites for ideas about accommodation, transfers (should be easy enough for 12 pers.), ski school etc.
Places I'd avoid - Bulgaria, Slovenia, Romania. There is a reason that they are cheap.