I played on the chess team at school for a couple of years and nobody had a problem with it. The competition night didn't conflict with meetings. I don't remember any conflicts about the cross issue either. Whether by coincidence or not, my parents bought a chess set that had figures rather than the standard shapes. There was no cross on the king.
How is chess war-like? If you engage in any game, you are trying to win. Even if you aren't "keeping score", you are trying to do better than somebody else. That's semi war-like. Yet all games aren't "bad".
Is it the eliminating of the other player's pieces? There are so many games that involve eliminating your player's pieces, or at least setting them back. Checkers, Sorry, and Backgammon are some that come to mind. It's an element of gaming.
Is it the names? The rook? The Knight? The king and queen? Strictly speaking, the pieces could be named anything. You could call them kittens, bunnies, and puppies if you'd like and still play the game with the same rules and moves.
There is a board, there are pieces, and each piece has specific moves. It's just a game. The moves and play are simple, yet it can become very complex. There is no element of chance. It's mind against mind. It's an awesome game.
Even though chess wasn't banned by our parents, games like Battleship and Stratego were. Monopoly wasn't. What's the lesson there? Greed? Accumulate as many worldly possessions and as much money as you can. We could play cards and yahtzee and other games of chance (the roll of the dice, the shuffle of the cards). Games that, at a young age left me believing in being lucky and unlucky and had be believing in superstitious things to make me more lucky.