Why is eye-based ray tracing preferred over light-based ray tracing?
As you point out there is problems about which pixel on the camera to connect to. You probably wouldn't notice where you took the sample on an area light because generally the light is pretty uniform throughout, and even if it is textured you can just take a few samples and it will average to look ~correct. However if you connect to a random pixel, you would very easily notice that some pixels are black or have aliasing problems. By starting at the camera it's easier to ensure that all pixels are being sampled enough.
You are also only considered the light from one bounce. If you also want to get indirect lighting then your statement about weights comes into play again, and you are more likely to get a good path by starting at the camera.
Finally there are many techniques which start from the light! And techniques which start at both ends and try and connect. Try searching for photon mapping and bi-directional path tracing. It just turns out that forward path tracers generally work the best in most situations. Renderman has an integrator which combines all these techniques  which you can read about here https://rmanwiki.pixar.com/display/REN/PxrVCM .