Scooterotica

They are pretty common in the classic race bike world. Much lighter and much smoother, making high speed carburation easier as there is less frothing in the float bowl, and less parts break and drop off.
For any sort of life, they must be set tight, once the teeth start to climb out of the pulleys, they will rip off. Alignment and tension is critical.
There was a lot of noise on Britbike about the dry/oilbath question, and other general desirable design features. A transmission design engineer and a maker/seller went at it for quite a while.
Not a lot of choice in Tony's application without lots of structural changes, but not all belts will survive with oil on them.
The belt is essentially inextensible via the kelvar/steel belts. They will not/do not provide any shock absorption for low speed impulses, but with the diameter of the front pully, some sort of cush device could be devised?
Chains have a fundamental flaw when used on small diameter sprockets as per bikes. As the roller engages it sits high up on the tooth, as it rotates it moves to the bottom, the changing working radius causes high frequency vibration.

I run a belt primary on a h-d its a Kevlar 3" belt feeding a dry clutch, it descent tend to be any quitter than the traditional chain and wet setup, It was fitted to allow for future power improvement/lack of power lost in the primary setup.
The primary runs directly (no cush drive) from the top end to the gearbox, and is supposedly good for 300bhp. It makes stripping the primary/drive sprocket easy meaning that the clutch can be stripped in minutes.
The set up on my bike is all dry, but there is a wet version kept within the standard cases.

Personally I cannot see the point of a belt primary on a Lambretta, parts would be expensive. Unless you sealed the gearbox in, on you'd have to run the belt/clutch wet, adding to complexity.
But I'm wiling to listen.