I imagine such a civilization could become quite advanced, but not easily - and not without some truly nasty tradeoffs.
So, there are ten year olds who are quite advanced, can think and plan at an adult level - but there are also ten year olds who are decidedly not. Given time, training, pressing need, and, eventually, a culture which expects them to fend for themselves, I suspect most ten year olds will be able to manage to survive themselves, and contribute to a working society. However, this will not be the case, initially. A lot of people would be lost in the wake of the disease, including a lot of children... and even after things settle down, the kids will be in survival mode for a long time as they try to survive, long before building up a new society. Getting back to the first point, those kids who aren't especially advanced (moreso emotionally than intellectually, this is about kids who can't or won't do their work, not those who can't solve complex problems) will likely not survive - and those who are, will have to let them die. It's a survival oriented equation, and kids who will survive and who lead groups who will survive are going to have to be ruthless - which will persist into the surviving societies.
In any case, one point I don't think you quite see is that, your kid-civilization is going to be, essentially, domesticating the adults. They are not going to have the resources to subsidize these unproductive mouths, and additionally, are not going to see any reason why they should for a very, very long time. If they have kids who are very clear headed, long thinking, and also really well educated about reproduction and genetics and diversity, minimum sustainable populations, and a host of other stuff, they might keep food out for feral adults, or once they have surplus, keep feeding people till sixteen or so (breeding them so they will have kids in the tribe's reach), and then letting them go. Even if that means letting the feral adults starve, or be prey to predators or each other. Well, they might use them as draft animals for heavy labor, or just killing them instead, I suppose - but the point is they will be treated like, and thought of like, domesticated animals.
For a long time, past the point of the kids forming self-sufficient tribes and up to the point of keeping track of sustainable numbers, the population will be surviving mostly by adopting the kids born of whatever feral adults may have survived. This means you should probably figure out how well these adults are surviving and/or capable of raising kids on nothing but instinct - after all, rabbits do survive quite handily, so it isn't impossible. Maybe the kids will keep track of, or temporarily care for pregnant women, if they know or somehow figure out they need more people in their groups, but they will not keep adults long term - they haven't the resources. And once they get to the point where they do know they need to keep the population up, they won't be caring for their adults, they will be breeding them for offspring - they will not consider them people anymore.
Also, it begs the question of why ten year olds, specifically. It would actually make a lot more sense to tie this to puberty, even if you want to have the average age of puberty, or some necessary pre-puberty changes which act as the trigger, happen around ten (I think ten is actually the lower bound, so minimal handwaving if you go this route). It can make a story-point whether the oldest are kicked out before they lose their minds, or after, or are kept for some time as breeders - which an odd several-year gap between losing their minds and reproducing. Also, if it gets tied to puberty, you might end up with an odd age creep going on, where those who are late bloomers survive much better. You likely won't buy more than a couple years this way, but it will help if you get even occasional twelve or fourteen year olds in your tribes.
So, your little-kid-tribes will be adopting little ones out of the wild - maybe toddler age, maybe finding abandoned infants, depending on how well your adults actually function. Little kids need care, and teaching (and this is also work). Your tribe will need people to manage the food supply - hunting, foraging, farming, they can all be options as your societies evolve. Kids will simply not be capable of the same amount of heavy labor an adult is, but with clever use of tools and draft animals (potentially including feral adults) they should be able to manage - and they will also not need as many resources as an adult would, including smaller dwellings, clothes, and meals.
The amount of time left over for innovation and advancement will be small, since even becoming more responsible younger and reaching social maturity early will not give the same kind of time-span to achieve things in, so society will advance slowly - though you might, maybe, get an artificial boost from leftover teachings or materials left over from the society before the disease.
And your society is going to be ruthless. Kids won't have the resources to be otherwise, especially in the immediate aftermath of the disease, and that will carry over into their civilizations. Also, kids tend to see the world in black and white - they won't have time to finish developing and get enough experience to be otherwise. So ruthless, they'll be hard on those unproductive, and unforgiving of defects (mental or physical). They will have to exile their adults, or otherwise move them from people-status to draft-animal-status - this will effect their empathy, they can't afford to be otherwise but they will also have little sympathy for those who won't, or can't, do their fair share - and will have practice at moving others from people-status to not-people-status, it will be easy for them to dehumanize others.