I'm pretty sure it's in the computer. Once you pass the time, they can dump you from the flight. I've heard GA's discussing this and they sometimes will take many factors into account to decide when they dump you to clear standbys or when they wait. ("He's GS, I'll give him a few more min", "He's local and still hasn't checked in? I don't think he's coming", to me: "I have to give him 1 more min to checkin, then I can give you the seat", etc). Either way, 100% your responsibility. Ofcourse you can complain to UA about anything and they're probably throw you at ETC eventually just to make you go away.

I'm curious what the exception was that didn't allow you to check in? . 99% of people (not flying basic economy) can just check in online so they don't really have the issue you speak of. Maybe they are delayed boarding ontime or getting to the baggage drop but not actually checking in. Ideally, you would have been able to checkin online but something prevented you from doing so.

What you describe at EWR doesn't seem too much different from a typical day. Poor line management, seemly only one person able to fix your problem.
This happened to me alot during covid trying to get my passport or test docs checked to be allowed to checkin. Even if I was already checked in and just wanted to print a boarding pass, same issue. There would be a huge line in the 1K/Polaris area and the guy would say that was the only area for intl flights. I'd ignore him and goto another area and find an agent to help clear me. The one agent didn't go in order of who's light started flashing first, she would just goto whoever was closed. Being more proactive/aggressive was required to get her attention. And that's just checkin, don't get me started about CLEAR.