I’m going to hazard a guess and say that with 10, cost IS an object. So you really need to prioritize needs versus “what if” fantasy. 4 sets of washer/dryers is probably not the most useful way to spend your limited budget, for example.
Is DH handy? If it were me, I’d home build triple bunks. Minimize the use of floor space that way. Make sure closet space is enormous so you don’t need dressers (at least 3.5 feet accessible width per kid in the room and use organizers to have shelves and two levels of rods). This will work for YOU as well as ensure the floor plan has market value many years down the road when it is your empty nest to sell. I like your instinct that what you need is well planned rooms, not lots of them. This also fits better for eventual resale. I wouldn’t try to give each kid a desk. Instead, plan work nooks in several places in the house (dining room, bedrooms, basement, etc).
Plan places for shoe storage. A massive mud room with room for coats, snowpants (climate dependent!), mittens, hats, shoes and backpacks would be awesome.
Laundry needs a bit of thought. Definately should not be on the main level. Either plan in an adequate space on the bedroom level or put in a chute all the way to the basement. Either one has advantages and drawbacks. The basement likley has a lot more room to hang dry things, iron, fold, etc at YOUR convenience, but its a lotta stairs. An upstairs laundry likely means cramped room, little space to hang dry, no permanent ironing spot and piles of clothes sit in your line of sight all day (argggh!). Less stair climbing though.
A garden shed seems like a godsend here too. Just all those bikes alone would eat half a two car garage! Throw in a lawn mower, tools, etc and where do you park? Outside I’d bet. Sheds are cheap compared to other stuff.
Got a decent basement that can be partially finished? (no water leaks or such problems) Basements make a great place to send noisy raucus kids to play and is a nice way to get the TV away from ‘in your face’ all the time. IMO, TV shouldn’t be front and center on the main level of the house. Sends a depressing message. Banish it to the basement where it SHOULD rank in family importance.
If doing a total rebuild, resist architect attempts to add soaring double height ceilings. These are echo chambers and will ruin MANY a nap. Not good for small kids and wasteful of space too. 9 foot main floor ceilings are a better way to a similar end.
Energy efficiency. If doing a total rebuild, take the opportunity to insulate well, get good windows (double pane, inert gas filled), 90%+ furnace, power exhaust water heater (or tankless) and maximum windows on the southern exposure (with foil lined cellular blinds for blocking summer heat). This will pay you back fairly quickly especially since the taxpayer (i.e. me and my future grandkids) is currently paying a portion of the cost.
Resist trendy and go with classic. Stainless is all the rage, but looks like dung with kiddy fingerprints all over it. Decline. White appliances are never trendy and never tacky. Architects will practically demand you get granite in the kitchen, but it can stain. Formica is still cheap, available in many styles and won’t give you a heart attack WHEN the kids some day damage it.
Laminate floors are trendy, but fragile - especially if you have boys. They don’t look too good when said boys puncture it by dropping iron frying pans and you can see the particle board underneath! (Luckily I learned this via in-laws, not my own kitchen). Wood still looks good even when dented. Ceramic similarly chips. Stay cheap with sheet linoleum or go expensive with real oak (actually not much more than ceramic).
Lighting. If things are going to be torn down to studs prewire for as much lighting as you can think of. Overhead lights, kitchen undercabinet lights, recessed kitchen ceiling lights, sconces etc. Even if you have to wait until later to actually buy the lights, prewiring is cheap and makes basic spaces look expensive and custom.
Our kids are used to sharing and we’d hate to split them up. They would be lonely!!
What a house! I think I’d even keep the grate if I could afford it 
