If you find chess variants interesting, I would recommend looking at Go [1] as well. It is similar to chess in that it is an ancient turn-based strategy game for two players and with perfect information, but I find it much more pleasing from a game-design perspective. The rules are much simpler, there are no draws, and the starting player does not have an advantage. Also, you can have handicaps so that two players with different playing strengths play a evenly-matched game.

To me, this shows that it is possible to solve most of the commonly cited issues in chess by changing it into a radically different game. But can you do it with only a minute adjustment? I find it difficult to imagine chess without draws, but it is not inconceivable. [2]

[1] https://en.wikipedia.org/wiki/Go_(game)

[2] E.g.: if the game would end in a draw, instead the player who has made more King-moves wins. If that is tied as well, Black wins.