If you were really trying to be minimalist, then the proper dice to use would be the d4, the d6, and the d10. The d8 is basically 2d4, and the d12 either 3d4 or 2d6 ( and of course d100 has always been modelled by most sane people with 2d10, instead of a real d100, ) .
But since no one is arguing for being that minimalist, I think it only behooves us to make use all of the classic dice. It would be a strange venture to try and include every legacy PHB race and class without including all of the dice.
It's actually easier to do a d4 with a d8 than the other way around. Same with a d12 and d6 and d20 and d10. All you have to do is half it, while to do the reverse you need to roll another die to see what multiple it is.
Indeed when D&D started, there wasn't a d10, so you had to use a d20 and just drop the first digit