Belts are slightly less efficient than chains; plus you either have to go SS/fixed (i.e. one gear only) or use an internal hub, which also reduces efficiency. Chains are also easier to repair, whereas a belt can only be replaced. If you opt for gears, any weight savings are more than offset by the increase in weight of an IH over derailleurs.
Ergo, you're compromising performance and one aspect of maintenance for a cleaner drive train. Although I'm sure better belts will be made moving forward, belt drives make sense for some applications but not for others. I doubt it will end up as the "wave of the future."