I just ran some monte carlo simulations. Turns out I was totally wrong, at least for a uniform distribution. I now suspect it's probably true for any distribution, but I can't prove it yet.

It just seems really odd to me: It seems like because you get 9x the number of dice rolls for one group you would end up with a final distribution that favors the majority group, but that is obviously not the case. I need to wrap my brain around that one.

I guess he is alluding to effects of rounding up/down in small samples but those cannot be biased in one direction. Nor are they certain to happen.

If I toss 9 quarters and a nickel and count the fraction of tails that rolled on the nickel then it will be equal to 1/10 of all the tails only if all coins roll tails or all roll heads i.e. in 2 outcomes out of 1024. In half of the other cases nickel tails will be overrepresented and in in the other half - underrepresented. It's because you cannot roll a fraction of a tail.