We have extremely hard water at home. So bad that I had to install a reverse osmosis system for our drinking water. Fed it to a faucet at the kitchen sink, as well as the fridge, so we get clean ice and water. It's amazing when you see the difference between a regular ice cube and an RO ice cube. You can almost see through the RO ones.
That said, every so often I have to clean literally everything, including outdoor spigots, with lime-a-way. While I hate suggesting a harsh chemical, it absolutely does work in breaking up the lime and calcium. You'd probably have to fill the tank with some, run it through the system a few times, then flush the entire system with bleach water. But that's just a suggestion. It's definitely a good idea to make sure it's not pipe shavings, as already suggested.
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