The idea is to create a sphere of about 1 kilogram and then both weight it and count the number of atoms in it. This is only possible by using crystalline matter, by taking advantage of the regular arrangement of the atoms.
Diamond would be indeed a perfect candidate but machining diamond is a hell of a lot more difficult than machining a crystal of silicon, because of the huge difference in hardness, a problem which pales in comparison with the sheer impossibility of making a diamond mono-crystal weighing one kilogram! The world record is about 20 grams. The difficulty is the need to apply a pressure of the order of 100,000 atmospheres, which is only possible in too small a volume for the target weight of 1 kilogram, which would be a cube with 6.5 cm sides. We could imagine settling for a bunch of smaller diamonds of course but this would introduce an extra source of uncertainty. Since it is possible to make a monocrystal of silicon weighing one kilogram, by using refinements of the growth methods developed and refined by the electronic industry, it would not make sense to consider diamond.
Why not graphite instead? Unlike diamond, it is possible to carefully make big enough mono-crystals. Unfortunately, graphite is made of a regular arrangement of carbon atoms strongly bonded in sheets, and those sheets do then stack up and keep together because of weaker forces between them: in particular they can easily shift with respect to each other. As a result, this makes graphite much less suitable for the described precision experiment where pinpointing the atomic arrangement is key.
With graphite and diamond, we have exhausted the crystalline phases of carbon. Thus exit carbon!