My recently acquired Pavoni had some sediment, too. After doing what you did, I filled the boiler about 1/3-1/2 full of water and then gently shook the machine, and dumped the water out. I then let it drain upside down in a dish drainer overnight. Don't forget to remove the grate on the drip tray first. (Don't ask how I know this). This seems to have taken care of the sediment problem.
The manual describes the "commissioning" procedure, which includes this step:
Prepare a water solution with two teaspoons
of bicarbonate in a container and wait for the
bicarbonate to dissolve completely; then pour
the solution into the boiler.
Alternatively, instead of using bicarbonate,
prepare in a container a solution of water with
a tablespoon of white vinegar and pour it into the boiler.
You then pull some blanks. You might want to try this as it might improve the taste. If you try this, you'll want to do the shake and drain afterward to clean out the boiler.
Final suggestion: have you tasted boiled Brita water? Maybe it's just not tasty stuff. I'm being serious, BTW. Brita water tastes bitter to me right out of the carafe, and I'm not sure that boiled water tastes good ever.