I'm going to leave side the agent's obvious issues.
If I read this correctly, it appears to complete your party's check in, you simply needed an employee's badge scan at a kiosk to confirm that your daughter was flying with you. And you missed your flights because there wasn't a UA employee available to do this. A dozen or so other passengers also missed that flight because regardless of reasons, UA ended up insufficiently staffing a busy counter. That seemed to me to be the crux of the issue.
Because it was so many people, the event must have been noticed. Seems to me your delay is an operational failure by the airline caused directly by their being understaffed.
Forgetting the IDB which doesn't apply as others have made clear and even leaving aside the agent's horrible performance, I'd be asking UA for reimbursement for hotel and meal costs. I doubt you'll be alone. I'd like to think they'd make good on it.
Fwiw, I have no appreciable status on UA and was treated fairly.
Also, iirc, when I've been delayed and missed a cxn on UA I got it at the layover airport, where the hotel voucher was needed. (Admittedly, this has only happened to me twice and it's been a few years.) It wasn't even a question. When we landed, missed cxns were directed by a gate agent to a desk where the vouchers were distributed. .