As to getting 2 different stories, I expect the 1st "they would make the flight" came from a roving staff person who was there to help people with check-in machines, which they declined to use, and the 2nd "the flight had ended the check-in period" came from the actual check-in staff. If they had checked in online, or even using the machines, there may have been a faster line for bag drop-offs.

Maybe they didn't pre-book their seats, and wanted a real person to get them seats together, but that may have been impossible if they were the last to check in to a full flight. And even if they had booked seats together, their seats may have been given away to passengers who were overbooked. I got caught in a situation (connecting from a partner airline that could not check me thru) where AC gave away my seat and gave me a standby boarding pass, let me board as standby, only to pull me back off the plane because it was overweight and I was only standby.