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Oh do I think it is possible for this to happen? Yes. But you truly do need to have a government which is willing to give up power and to have faith in their system. You can't think you're the only person who can do your job, or that you're the only hope for the development of the country.
Typically rulers in these countries lead for decades, because they think they're the only hope for this country. When in reality they're not.
Without intervention, it'll be a slow climb. As the people's lives improves and become economically-free, they'd start asking for political-freedom, with more say on decision-making.
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Unfortunately DR Congo and Botswana have very different last 100-120 years, even though they were both colonies. DR Congo history is way more violent with longer history of exploitation, and way less democratic. As @squatz1 mentioned, their presidents tend to rule until they die, while Botswana has longest streak of democratic elections in Africa.
Botswana was once as poor as DR Congo, but their political stability helped them to use their natural sources properly, and until that is improved in DR Congo, there won't be much progress, no matter what kind of natural resource they find there.
Regarding children working in those mines, some are definitely forced to do that, but for some that is the only way to get money for food. I read some interview recently with 11 yo that is mining cobalt. His parents are dead, and all he has is his grandma, and it is up to him to provide food. Kids there don't really have much choice, eventually they could be recruited as child soldiers, as that is still going on there. Sadly, there are worse things than Cobalt mining for kids there...
Most article I've read chalk it up to luck and determination, with a group Tswana chieftains even going so far as to travel to England to set-up proper diplomatic ties. So yes, they managed to secure political stability and that has helped them weather the diamond discovery.
Currently, I think it'll be better for the government to grow big enough first to stamp down armed groups which is hurting the stability of the country (these are those that hire children as child soldiers). Then the political freedom for the people will come later as more of them become affluent enough. Of course some people would think it should be the reverse but that's just my opinion.