I think chess teaches a lot more. It teaches decision making and resource management. It's very important to consider your options and decide how much time you spend analyzing them. It teaches you things like elimination process (if you have 3 options, one complicated one and 2 simple ones then start analyzing simple ones because if they are bad you just saved a lot of time by not analyzing 3rd one).

It also teaches you to forget about the past and focus on the present. It teaches you long term plans usually don't work and relying on them is a bad idea and it's better to focus on execution here and now (this is what Soviet school got wrong although probably only officially, the players knew it all along - analogies to 5 years plans were just too tempting for book authors).

At human level it also teaches you to design strategy to exploit your opponents - play openings they hate, simplify vs players who like complicated positions, complicate vs players who like simple ones.

There is a lot of interesting things you learn about thought process and decision making as your level improve. It might not be the best game for that but it's pretty good considered how simple it is.