Summary: OP does exactly what DL asks him to do, even checking in online at home and arriving an extra 30 minutes ahead of time at the airport. DL understaffs and equipment malfunctions. OP pleads his case with DL staff on site, in real time, and is rebuffed despite 3 children in tow. He is penalized $600 for his compliance with the rules and told to take a hike by DL corporate. DL apologists say he should have known better, and now he's learned his lesson!

Boo yah! How rich!

OP, I like all of the following suggestions, in no particular order:
- file complaint with DOT
- file compliant with AG
- contest charges on credit card
- sue in small claims court
- write a song about DL's incompetence and publish on Youtube
- start a blog about how you "hate Delta"
- find another airport
- find another airline

Definitely keep us posted. You were cheated.