Deep Blue did not "solve" chess - it and the generations of computer chess programs since have demonstrated they computer programs have gotten to grandmaster level (or just beyond) in playing chess - BUT - they have not "solved" chess.
Solving chess would imply knowing who will win from the initial starting position (i.e. showing a forced win for either white or black from the opening).
And more recently it has been shown that an average chess player in partnership with a modern chess program can beat grandmasters by themselves or the best of modern computer programs (i.e. human+machine is better than human or machine alone).
Chess is a game that changes as you get better and more studied in playing the game. One of the most important books I've ever read was Emanuel Lasker's Manual of Chess (published now quite a long time ago yet still a classic and great book). Emanuel Lasker was one of the best chess players in the history of the game - was the world champion for many years - and he was also a philosopher. His manual of chess teaches chess in the opposite manner of most teachers (then and now) - he starts with the endgame and works up to the opening.
But more than his chess lessons (which are fantastic and still compelling today) he was also teaching a way of thinking - a way of looking at the future, as your own decisions and the impact of them on others (and of other's decisions on you). He deemphasized the memorization that is all to often how chess is taught in favor of teaching you to evaluate positions, make plans and understand situations. Learning to think for yourself, to modify your plans when the situation changes and learning how to evaluate honestly your position (and that of your opponents) is a skill which carries over into everything I do with my career.
Further I disagree that "any" modern game can teach the same lessons as chess.
Many modern games, unlike chess, involve randomness as an aspect of the game. Learning to deal with randomness (and to understand probabilities etc) are important lessons but are not part of chess.
Unlike most modern games Chess is not a win/lose game. Draws are core to chess (though frequently poorly understood by beginners). Learning to recognize a drawn position - and when to offer a draw (and when to accept one) is a lesson that few games other than Chess offer.
The literature and history of chess as well as the universality of chess are fairly unique. Few other games have as rich a history or as deep a library (poker today may be beginning to come close, Bridge has some and Go has others - but Go for example is not a global game to the same degree as Chess). I've played chess against opponents with whom I shared no common language, something hard to do with most other games.