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Then what happens if one child who knows they won’t get anything, refuses to provide their piece?

Don’t give that child a piece in the first place!

Even worse, what if one or more children lose their bits and can’t provide their piece?

Have each child make horcuxes of their horcrux :)

Not all K of N schemes are "secret sharing" (the most well known ones aren't), though I agree that for this will case you probably want a secret sharing algorithm and not say, an erasure code (as you want the property of zero information transfer); however, certainly not all secret sharing algorithms are Shamir's specific secret sharing algorithm.

This is what is used in hashicorp vault

And in TFA