Why is chess considered special for educating a mindful person?
Chess is considered "special" due to certain attributes of the game. From "Beyond the 64 Squares: Does Chess Instruction Enhance Children’s Academic and Cognitive Skills? A Meta-Analysis":


  Two main explanations have been adduced to support this hypothesis.
  First, chess requires decision-making skills and high-level processes
  (such as acquiring and selecting relevant information from a problem)
  similar to those used in mathematics and reading (Margulies, 1992).
  Second, since chess is a demanding task involving focused attention
  and problem solving, playing chess should strengthen these cognitive
  abilities and thus be beneficial for children's school performance
  (Bart, 2014).


However, this consideration is only marginally valid. In "Enhancing Creativity in Children by Imparting Chess Training" the increase in mathematical skills from Chess was compared to mathematical instruction and playing the game Go. Although Chess outperformed Go, thus cementing it's "specialness", it did not beat basic mathematical instruction.

In conclusion, Chess is a bit "special" among games in terms of cognitive improvement, however it doesn't beat actually doing a task and it seems unlikely for any other game would be superior.