Although passengers from fee-requiring countries really shouldn't be getting successfully onto a flight to Argentina without proof of payment, it does seem like something that should be able to be sorted out in cooperation with the arriving carrier on arrival... after all, payment is taken online, and proof is a printout, so all that should be needed is access to an office and a credit card.

As an airline can be fined (and/or have to incur cost to carry a passenger back) in cases where entry is refused and documents were not checked prior to accepting the passenger, I would think any carrier with such a passenger on their manifest would find it in their interest to get them to a transit passenger area, sort out payment of the fee, and eventually get them on their way.