All the paperwork and calculations are done once the flight is closed to check-in. Technically, you can accept passengers after check-in closes, and the paperwork can be modified (Last Minute Changes), but the main point is consistency.
First, if a passenger turns up late with a bag that is not fit for hand-luggage, there is a high risk that the bag won`t make it to the plane. But by accepting the bag, the company is accepting responsibility for it, so if the bag is not delivered at the other end (very likely to happen at this point), the company has to re-unite the bag with the passenger, at the company's expense.
Then, consistency. It says in the ticket that check-in wil close x minutes before departure. When do you draw the line? What is the maximum time you are going to accept a passenger? And if you did it last week, how come you are not doing it today for me? Itīs all down to set a time framework and adhere to it to ensure a good service, ie not delaying the departure of the flight.
Once check-in is closed, all departments work with those figures. Load control issues the loading plan and the loadsheet, and trims the aircraft for optimum performance and safety, catering confirms the amount of meals (in full services airlines, of course), check-in coordination accepts whoever is on standby basing on the amount of available seats (in full flights, the seats of those passengers that haven't checked in when the flight closes), and lots of other things. Whoever is responsible for dispatching the flight works with those figures and issue all the paperwork (which, by law, has to be strictly accurate) for the flight. They can add last minutes changes, but that can affect many things. It is up to them to decide whether accepting a pax that has arrived late would affect the operation. Many factors are involved. Have they got time to make it to the gate? Is there a long queue at the security control? If you accept a passenger for a flight, you are considering that the passenger can make it to the aircraft, so if they don't, it's the airline responsibility. Companies used to have last minute passengers "on chance only". That meant that if the passenger doesn't make it, the company declines responsibility, and it's the pax fault, but in practice, you have accepted them and you have denied boarding.
So, to avoid all this, and standarise the departing procedure (an aircraft departure is not like a bus departure), companies are enforcing the check-in closing times strictly. Itīs a clear part of the contract and it's the passengers' responsibility to be on time. The different departments start working with those figures as soon as the flight is closed, and any last minute changes can affect the operation, as well as creating an eternal inconsistency in the way the terms of the contract are applied, with the associated passengers' perception of "if they did it last time and they are not doing it this time, it's because all these excuses for closing the flights on time are bs!"