Besides the rules themselves, knowing the OP's origin and destination might answer one key question - why no offer of an alternative flight besides the next morning? The timing suggests a TATL flight, wondering if it was from somewhere that had no other reasonable options? Otherwise, I would think DL's obligation would've been to find OP an alternative flight plan to get to the destination ASAP.

I completely understand why the OP is angry - missing the interview is tough, and it's not fair to assume the OP could have done anything about it with better advance planning without knowing the circumstances.