Running water causes a lot more erosion than stationary bodies of water. Consider lakes, which are still cycling water much like a river, but over thousands of years they deposit so much silt that they cease to exist. That’s the opposite of erosion.
Underwater erosion is certainly a thing, but in comparison to downhill water erosion on land, it’s pretty insignificant. It does not seem a given that it could significantly offset the processes that remove salt from salt water.
Yeah, but not all that salt is in contact with water. There are huge deposits of salt under the Mediterranean, yet they have not dissolved into the ocean. You have to actually strip away the non-salt components to get enough water to salt contact to actually dissolve enough salt to make that much water salty.