I completed the training, go to the menu and select story mode. I create a save, click on it and then “play” aaand... nothing.

My game menu simply restarts after each click on "play".

Try "regenerate data.sav" in settings. If that doesn't work - Go to your main drive > Users > Your PC name > AppData (click view at the top and tick Hidden Items if you can't see it) > Local > VotV > Saved > SaveGames and delete everything inside. Now re-launch the game and regenerate data.sav. If THAT doesn't work, then it's possible the game didn't extract properly.