There are new approaches to generating realistic character motion being shown at SIGGRAPH every year, getting better and better, but the best ones are largely too expensive for games to adopt. Plus, for player-controlled characters, at a certain point you run into a fundamental tension between control responsiveness and physical realism of animations.
I recently saw a great article on the topic which even comes with a WebGL demo: https://theorangeduck.com/page/code-vs-data-driven-displacem...
