Need a rule clarified
Runner on 1st no outs, batter swings and misses 3rd strike, catcher bobbles ball, batter takes off for first gets catcher to throw to first, runner on 1st takes off for second, ball is over thrown to first, runner ends up on third, batter is out , coach argues that the batter made the catcher throw to first even though he is out, wants the runner to either advance 1 base or go back to first, explained that runner advances at own risk to be put out and batter is automatically out and the catcher did not have to throw to first, what is the rule and was the call right