Diameter and distance (focal length) actually come into it.
I'd say the ratio (diameter)/(focal length) is what matters for the efficiency. The LED has a tiny lens by itself, but the cone of light is what, around 45+45 degrees so 90 degrees total? Probably even wider. That means that to catch all that light, the lens has to be twice as wide as the focal length.
The f-number of a lens (like you have a bright lens on a camera, f 2.0, or a not so bright f5.6) is defined as (focal length)/(diameter). This means you need an f 0.5 lens to catch all light. Oops.
Now, it is not only about catching light, but also about focusing. The longer the focal length, the narrower the beam will be. But with the lens diameter equal, it means less light to enter the lens. So let's see:
- double the focal length -> spot is twice as narrow, 1/4 the area.
- double the distance from the LED means catching a quarter of the light (within reason as the LED is a bit focused already).
So, a longer focal lenght with the same diameter means a tighter throw but one that is not brighter.
To look at the bright side:
- The diameter determines the throw beam center brightness. If you need throw, diameter is it (actually a smaller/more concentrated LED chip helps too).
- A shorter focal length (thick lens, close to the LED) means the throw beam is wider (but still as bright).
Do throwers exist that use dial optics, with an extra, fixed, tiny focusing lends on the LED to direct all light into the big moving lens? That would definitely help!