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The joints keep flaking out or absorbing stain. How can you prevent this?
Only way to combat that is to wipe down with a kitchen towel after each use and clean regularly.
I bet that if your kitchen faucet spout were to discharge water beyond your sink's midpoint, or directly over the strainer, you would have less splashing.
If the sprayer is used lower in the sink, with less water pressure,
splashing is reduced.
You may want to apply a heavy layer of car wax behind the sink. Wipe dry after using sink, as suggested. Clean regularly with a rag soaked in distilled white vinegar, so long as your surfaces are not stone.
Use a grout sealer on the grout. Spray some plain canola cooking oil on a cloth and rub it on the tile. Will repel the water.
Maybe the towel is the problem. I would think it would keep the area damp. I think your idea of clear silicon is a good one. Fill the entire area with it slopping out to the sides. Also there is a caulk that can be applied in wet conditions, even outdoors in rain.