In their FAQ they say they're working on MTB wheels, but that the challenge is the (lack of) lateral stiffness in a bigger wheel, which makes sense. The larger the diameter the spring needs to cross, the harder it is to have it only flex in one dimension.

I would imagine weight - especially for road bike tyres.

For comparison, a lightweight Zipp wheelset comes in a just over 1Kg. A mid range lightweight set will be in the 1.4Kg range.

There are other ways to make a road bike handle bumps and road buzz, e.g. specifically constructed carbon seat post, or flexible seat stays.

A good race to watch for these kind of things is the Paris Roubaix, some of the bikes had rubber inserts to the seat stays a few years ago, but now have enough damping just from carbon layup of that section.

Aerodynamics would be horrible, on road bikes that would be a first order downside.

As well, tires are already integral suspension and often enough for bike applications.