Obviously a great deal has changed since then, but my parents did this with me and my two brothers back in the late 70s. I was 11 at the time.
We travelled for six months, in a camper van, through pretty much all of western europe. Well, Yugoslavia as well.
We stayed in campsites, and our education consisted of worksheets that covered only English and mathematics.
Education-wise, I got back to school at the end of the year just in time for the end of year tests, passed everything with no problems, as did my brothers, and started high school the following year with no academic problems and tons more general knowledge than my classmates.
Experience-wise, do it!
Obviously I was a kid, so won't have remembered everything, but a heck of a lot of the memories are crystal clear.
The memories that stand out the most were actually everyday sorts of things which differed radically from what I had known up till then. So, seeing a maypole for the first time or things like kids standing at the side of the road in Yugoslavia trying to sell just 2 or 3 figs to passing tourists or out working as shepherds. Or old women in Greece leading donkeys utterly laden down with straw or firewood along the road while all the menfolk seemed to be relaxing in the local cafe, etc. :-(
It's not just the initial experience, though, but also the ability to decades later revisit somewhere, whether in person or via the news, and have a memory and some sort of connection with that place.
Maybe this all sounds a little twee or self-centered but, for example, the war in the Balkans at the end of last century felt a lot 'closer to home' knowing that the people killing and being killed there were the ones/just like the ones I had seen their earlier, in the same terrain.
Or when you see pictures of a glacier that you walked on as a kid, with a staircase leading down to the surface of the glacier that you know you walked down. Except that the glacier has now shrunk so much that the staircase ends meters above the ice...
I guess I'm saying that for me, at least, it increased my understanding of the world and where one sits in it ...
Two downsides to note, though:
Spending so much time in such close proximity can bring you together as a family, but can also drive you rather nuts at times. Build in potential escapes from one another, even if its just taking the kids to playgrounds or beaches or whst have you occasionally.
Long term, your kids might end up longing for one of the places you visited and spend their adult life somewhere very very far from you.