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The reason water is concentrated in oceans isn't specifically due to continents existing. Salt doesn't evaporate so all rain is fresh water. That fresh water falls. When it falls over land it flows to the lowest point it can go. This leads to all flowing water flowing towards oceans and seas. Salt won't travel upstream. Ergo salt simply stays in oceans and seas.
Now consider a world with no land. This wouldn't really differ from a single ocean on earth. Currents and waves will move in all directions at some point which should mix the salt all around. You could get some differences if there were ice caps or icebergs. Those could behave similarly to continents depending on size.
Sure, I get that, but without land for rivers to essentially mine salt from, the equation changes a lot. Underwater erosion is dramatically less destructive than above water erosion.
Earth's oceans are in a steady state, where all the addition of salt by rivers is balanced by loss of salt in the ocean. If you removed all the rivers from the equation, Earth's oceans would find a new balance at a point significantly less salty than they are currently. Though I have little idea if that would be something we consider freshwater, or just "less salty" saltwater.
But that's not what's happening. There's no "mining" of salt. There's no significant addition or loss of salt to the ocean. Salt just stays in the oceans here. Freshwater will evaporate and return through rivers and rain. On a planet without land, the salt would still remain in the ocean.
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.
Ah. I see the angle you're coming from. I had mentioned in another comment somewhere that essentially all salt without an impermeable barrier between it and the water on this planet would be dissolved (provided it doesn't saturate the water which would be a horrifically enormous amount of salt). Salt is highly soluble in water and on any timescale that could be relevant would fully dissolve and achieve a general equilibrium. If the planet has water, then it has a star able to warm the planet. There's no realistic scenario that wouldn't result in the ocean fully mixing.