Three thoughts:
==
In hindsight, here is one thing you could have done:
(1) Book a refundable ticket for an 18 year old with your daughter's name out of ROC (any destination, any flight same day departing after yours). Check in online.
(2) Transit security with your family's boarding passes, including the unusual one for your daughter.
(3) After security, cancel and refund that ticket (UA rules say within 7 days this may be nonrefundable but they do not yet enforce this rule).
(4) Visit the boarding gate for your flight and ask the gate agent to make sure your daughter is checked in for the same flight as the rest of you. You're sorry, but travel with kids is hectic and you don't have the boarding pass. You're not sure the check-in agent got everything right and just want to double check.
(5) Hope the check-in agent isn't also the person handling boarding for your flight.
This would certainly constitute a violation of United's contract of carriage on your part, but since they were kinda doing that at the check-in desk too by failing to staff their desk with a competent agent, it would satisfy the time honored playground principle of "I'm rubber, you're glue".
==
Per your description of the situation, you were not involuntarily denied boarding under the ordinary meaning of the phrase. The obvious next step from United's point of view is to lobby the DOT for an official definition of "bad customer experience" which excludes your situation, so that we can say with certainty that you did not have a bad customer experience.
That would certainly be the simplest, cheapest investment -- just one lobbyist -- that could greatly reduce the number of bad customer experiences people have.
==
There are at least two schools of thought on how to respond to this situation.
One is that you should really give United's customer service team an opportunity to make things right; document your damages and ask them to make you whole.
Another is that you should immediately escalate via a DOT complaint, which will get more senior UA CS eyes on your complaint but will also generate a more final response (you had better be very sure of what you are asking for). Empirically I have found in similar situations that when United agents "incompetently deny boarding" and a DOT complaint is filed, UA sometimes voluntarily chooses to offer customer service compensation as though the situation were an IDB; that compensation strategy may not actually pay for your damages.
There is another school of thought that involves publicity (i.e. publish your 17-second video to /r/PublicFreakout and respond to reddit users' questions with your story), but it's very hard to get that right and I think unfair to the employee. Sure, you missed your flight, but what if they were having a major mental health crisis -- would you really want one bad day to ruin someone's career?