I am not being harsh. If the agent makes an exception and something goes wrong they will probably be fired. The cutoff is to allow for the airline to send info to the U.S government.
I am willing to wager that the operation in Cartegenia is not staffed by Spirit employees but by a private contractor who will fire an employee who breaks a rule. Remember this is a foreign country where employees do not have the rights enjoyed by U.S workers.

This is the rule from Spirits website "
Check-In

Check-in begins 3 hours prior to departure at the Spirit Airlines ticket counter.

Spirit reserves the right to cancel the reservation or seat assignment of any customer who does not have a boarding pass at least 60 minutes prior to departure for all international flights.

Why does the OP expect them not to honor the terms of the agreement.

As I said I have been late getting to the airport for an international flight. IT WAS MY FAULT and the airline denied me boarding because I WAS LATE.
This is called taking responsibility for my actions and accepting the consequences of my error.

The op was late and denied boarding a specified in Spirits T&C so I do not see anything wrong on Spirits side however the OP expecting to have rules waived for them and maybe inconveniencing everyone who followed the rules and is on the plane is wrong.
Maybe no one would of been inconvenienced this time but breaking a rule may inconvenience those who followed the rules and that should never happen.