This is a problem I am thinking about as well! My wife and I only have our first so far (9 months old) but we are looking to have 3-4 kids total.
Here are some of the thoughts I have had:
-Normal class A/C motorhome: lots of room but typically passengers sit sideways or lap belts only, so safety is a concern; large size means hard to drive, park, go down trails, and are banned at some parks.
-Class B motorhome: I have never seen one with seating for more than 4 from the standard factories
-Camper Van: if you include built in cooking gear and storage, it becomes cramped to sleep more than 4 inside once the kids start getting taller, but could be a good for the next few years. Maybe a good "5 year" solution rather than a forever camper as my wife would put it.
-Van: like others have mentioned, if you took a full size van, put a pop top, and had nothing but a single row of 3 seats, you would have plenty of sleeping space for the kids with mom and dad upstairs. You might even be able to pull off bunk beds if you did it right (some van outfitters sell bunk beds for sprinters but could probably work on others). Bunk beds can attach to the sides and fold up for more room for bikes and such.
-Tow rig with trailer: harder to do trails unless you go with base camp style camping. Some trailers are very rugged but most of those are built either just to carry gear, limited to 4 sleepers, or very expensive
The other option, go with something crazy and classic, but they tend to be slow, old (requiring more maintenance), and parts can be harder to find. Cool example below that sleeps 4+ if kids are small and parents can fit in the pop-top. Cool thing is some of those old vans have 4WD with low range. The HiAce seems to be the only one capable of US highway speeds, but since they were never sold in the states you must get something older than 25 years old (import laws), and they almost always come out of Japan so right hand drive. Right hand isnt hard, but some people think it is. I had a right hand drive before with a manual and honestly shifting with the left hand makes more scene as a right hander and is easier to drive than a left hand drive with a stick.
Toyota HiAce Cruising Cabin
Normal Toyota HiAce Super Custom
The more I think about it for my family, the more I lean to either a tent solution or a towed base camp solution. If you did either, you could go several different ways from a 4x4 van to an older suburban with the 3rd row put down for lots of gear hauling ability. Put a roof rack full of tents and sleeping bags and you could carry a lot of gear for not much money while still being able to do a lot of basic forest roads. The link below is a long detailed thread about a fellow expedition portal member that with very limited budge but some time has created a very capable 1500 series Suburban on 37 inch tires. He wheels it hard in Death Valley and has lots of great info to share
2000 Suburban K1500 budget low lift with 37"s