I can't see how anyone can think "the exporters pay the tariff" makes any sense. TBH, we'll never know how many people thought it made sense because it didn't matter.

In the end money move around. If - for example - the government would just give the citizens the money from the tariffs in equal share (I mean not that I suggest they would, but technically possible), it would be like taking from the citizens that consume more and give it to the citizens that consume less.

So, yes, it is correct in a practical immediate sense that "the exporters pay the tariff" but that excludes many relevant issues like how prices evolve (which are paid by the consumers), what the government does with the money (it could share or not) and what others decide to produce (to avoid tariffs). But definitely many people didn't thought of all that ...