Because the SI specifies the kilogram as the base unit of mass.

If you want to generate a kilogram to a certain precision (say, a part in 10^8) as a sum of a thousand well-measured grams, the grams need to be measured sqrt(1000) better (about a factor of 30). If you could multiply, you're fine, but since you're required to make (or measure) an actual kilo, you have to sum.

If we were to accept the multiplication method, it would be equivalent to restandardizing the SI upon the gram. That would be an okay thing to do, but to maintain continuity with past measurements (referenced to the IPK), we'd still have to make or measure an actual kilo, requiring precision sufficient to do the sum.

I don't think the problem is that we can't get anything that big to be precise, but rather that we just don't have any mass experiments that are precise enough period. If you look at the definition of a second, you can see that scientists aren't squeamish at all about multiplying things.

Because that maintains the imprecision. Multiplying won't somehow scale the precision up as you go. It will in fact magnify the imprecision.