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final part
Ideally, perfectly fair games are the goal, as the player's skills
determine the outcome. Practically and realistically, only a very small
percentage of all games will be fair, or better, so minimally unfair that
it is not noticeable [or, hopefully, measurable]. There are a number of
ways to achieve this, and the more ways that can be put into 1 game, the
more likely it is to approach the ideal. Symmetry in some form is an
obvious way of balancing a 2-player game, though it is not sufficient for
the task, and needs to be augmented with other methods. And a certain
imbalance can make the game more interesting to us 'sophisticates' [and,
if we play any of these things, we are at least gaming sophisticates] - to
win as black means more.
And, as I asked Derek, I'll ask you, George, if you wish, to comment on
Chieftain Chess, partly for the symmetry of the situation, and partly from
curiosity. I suspect you will have problems with at least some aspects of
it, as it has a total starting density of 33%, and only 5 types of pieces
on a 192-square board. 
Finally, does anyone else have an opinion on the original question that
started this topic: 'I wonder what is effect of symmetry of starting
setup on strategy. Comparing Shantranj and Chuturanga, it occured to me
that one has radial symmetry, while the other billateral symmetry. Which
one has better balance?' - Jianying Ji