This is all very interesting and pertinent. I was wondering about the hadean period, and whether you could actually get to an ocean world without first having continents with a water cycle. I don't know enough about planetary formation to conclude further. Thanks for pointing me to the hadean period, I will read more about that.
You might misunderstand my comment about the dead sea. The dead sea actually precipitates salt crystals onto the bottom of the sea. No land is required in this strange process. I don't think it's clear to say whether this happens because of the extreme salinity of the dead sea, or if the extreme salinity just makes it the only place we observe this rapid desalination on human time scales. I offered this as perhaps the most striking example that salts dissolved in water are not necessarily a stable state on a timeline of billions of years.
Salt gets into rivers when material that can't be dissolved is stripped away by erosion. This exposes new water soluble compounds to the water, where they dissolve into the water and are taken to the ocean.
Over millions of years erosion removes innumerable tons of material, essentially mining the subsurface soluble compounds and delivering them to the ocean. Once there, as you mention ,those salts remain in the ocean. On Earth, this process began billions of years ago and has been adding salt to the oceans ever since.
You can observe this happening in many rivers today. The Colorado River is a great one. If you measure is salinity at the headwaters (or heck, probably even the inlet of Lake Powell), and where it enters the Gulf of Mexico, you will observe an incredible increase in salt. There was an international treaty formed around the US delivering river water that is not too salty to grow crops in to Mexico. The US solved that problem by installing a desalination plant on the river!
However without that land based salt mining process, how salty would the oceans be? Lots of good clues in this thread, but I don't think anyone has offered a definitive answer.