The problem is not just to find another _method_ to predict game results, but to construct and evaluate a better workable scientific model of chess ability. That's hard, because the criterion 'game result' itself possibly is not a valid indicator of the quality of game play, and the stability of playing strength over time, which is reliability. To estimate these criteria, it is necessary, to design the data collection, as scientists do e.g. in experimental design. In addition, the available tests of logistic models, like ELO, are not sufficient.
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The problem is not just to find another _method_ to predict game results, but to construct and evaluate a better workable scientific model of chess ability. That's hard, because the criterion 'game result' itself possibly is not a valid indicator of the quality of game play, and the stability of playing strength over time, which is reliability. To estimate these criteria, it is necessary, to design the data collection, as scientists do e.g. in experimental design.
In addition, the available tests of logistic models, like ELO, are not sufficient.
Thus you can get answers for all your question. - Ok, it needs some expertise in the art of understanding: empathy ;-)