I think the only first world people having any issues with salt consumption are folks trying to live entirely off the land, and vegetarian leaning. While looking for other things, I've found a couple of videos talking about what plants bioaccumulate salt and how to extract it. I think it depends on whether the plant is otherwise poisonous. If it's edible then clearly making broth out of it would be the most productive option.

Many of our biggest civilizations have access to the ocean or river deltas. They would have had access to salt. From what I understand, predators get their entire salt intake from the animals they eat (blood, in particular?), and when we ate the entire animal that probably would have worked for us as well.

If you're trying to get salt from plants it's because you don't or won't have other options. There was a time that discovering salt deposits within your kingdom had huge political implications.