> They explain why this makes for a bit fairer dice here.

Well... they explain why the opposite-side numbering convention makes for fairer dice. They don't say anything at all about balanced vertex sums other than "we believe it's important".

Just compare the reasoning:

> If a die is unintentionally oblate (slightly flattened on opposing sides), the flatter regions are more likely to turn up when the die is tossed. If these two opposite numbers were 19 and 20 for example, then the die would on average roll high, since these two numbers would come up too often. Having the two numbers add to 21 avoids any such bias in the average number rolled. For this reason, the opposite-side numbering convention improves fairness.

vs

> Equally important in our opinion is balancing of the vertex sums.

This does not give me confidence that the vertex sums matter.