You are quite correct that an asymetrical game is much harder to balance.
However having identical sides and a symmetric playing field doesn't always guarantee a balanced game. For example, if one piece or position dominates all others it can lead to a lack of viable options and just one way to play, making the game uninteresting. You don't just want the players to have equal strength, you also want the universe of possible playing strategies to contain many different strong options.
Actually it has had balance changes. Chess clock for instance is a balance update between the players, but there's also been balancing between pieces. En passant and castling but also changing how the pieces work (for example bishop).
Despite the obvious symmetry of the game there's still a lot to balance.
Yes and this isn't necessary because the two sides are completely identical. No differences in pieces or terrain or anything so there's no need to change a piece to make it stronger or weaker.