Why is the violin played with the left hand doing the complex fingering and the right only bowing, when most players are right-handed and surely ought to find the opposite way easier ?

Fred, I don't play the violin, nor do I know the answer to your very interesting question. But here's my hunch. Just assuming the pose of playing the violin in pantomime, it seems to be more awkward to do the bowing than the fingering. My theory is that righthandedness doesn't seem to affect the fingers to the extent that it does the whole arm. Typing with both hands is pretty equal in my case even though I would find it impossible to write longhand. I fully expect to be shot down.

I play guitar and the same thing holds true for guitar as for the violin.The most difficult fingering is done by the left hand while the right hand strums or plucks the correct string or strings.While it would seem that the left hand has the more difficult job,it's the right hand that has job of selecting the proper string [single note]or strings[chords] to play .
I don't play violin but I'm quite sure that the angle of the bow against the strings will determine which precise string will sound.Bow tip up for the higher strings,flatter for the middle strings and lowered for the lower sounding strings.
If you want a more precise answer try posting it in the music section of Answerpool.You may get a better answer.