We've been sailing since our children were 5 and 7. They joined us on many a cruise too. There is no way I'd sail again on HAL and I certainly wouldn't let my grandchild either. Suite passengers have a dedicated lounge (Neptune) and I can guarantee you that your children would not be warmly received by the passengers using the facility.
Celebrity does have some wonderful suites. Your choice of attire could be easily accommodated. Suite passengers have access, on a space available basis, to Blu. On each ship Blu is a smaller more intimate restaurant that is always available for breakfast. Very few suite passengers complain about waiting a few minutes for a table at peak times. The dress code for Blu is always country club casual. Each Celebrity ship also has specialty restaurants. Depending on the suite class you could receive two gratis dining experiences in the specialty restaurants. Celebrity has a good children's program. It's not over the top like some lines but I love watching the youngsters parade around the ship and enjoy the pool.
The primo lines are really not geared to children. I'm sure the children would be well taken care of, and I truly think the passengers would be kinder than those on HAL. Our young cousins, now teens, have been on Crystal many times but they are very self sufficient and would rather take shore excursions than participate in a child/teen program. Their parents decided that Crystal was the best fit as opposed to Seabourn or Silverseas. They always sail in suites.
I haven't tried NCL or MSC but the suite accommodations appear to meet your criteria for lux style. It's just that the rest of the ship probably won't.
You probably wouldn't be too interested in a New England/Canada cruise but many lines start that itinerary in August and September, perhaps before the children return to school.
Suite accommodations do sell quickly so you may be challenged for this summer. Do visit a travel agent who specializes in cruises. They often can find "invisible" inventory.