No! Distilled water by itself is corrosive to metal parts, it has so little mineralization in it that it will actually leech metals from the pipes and boiler INTO the water. If you go distilled, you need to re-mineralize with Third-Wave Water or one of the DIY recipes for reconstituting properly balanced water.
The Rocket softener pillow/sheet is only good for a few weeks or months, and doesn't filter all of the water in the reservoir -- just what passes "through" the filter.
The triple-filtered faucet .... what does it filter? Usually that's "taste, chlorine, and mechanical" -- aka carbon filtration and a micron sediment filter. It will do NOTHING for dissolved calcium in the water, aka total hardness.
You have a few options.
Option #1 - Get a water softener cartridge and run your water through it. It doesn't have to be plumbed in, you can simple hook one to a faucet and pour out a pitcher of water at a time. Something like this:
https://www.amazon.com/Pentek-WS-10-Softener-Cartridge-155319-43/dp/B002711V44/ref=sr_1_6?dchild=1
Option #2 - Get a proper hardness filtering pitcher, like the BWT Penguin. Be advised the cartridges are expensive, and they have a very small capacity.
https://www.wholelattelove.com/products/bwt-penguin-2-7-l-water-pitcher
Ultimately, you will need to -regularly- TEST your water to be sure you are under 35-50ppm total hardness if you want to prevent scale build-up. Regular testing will confirm if your softening method is still effective or needs to be replaced/recharged.
There are cheap test strips on Amazon that do this, but you're better off with a liquid titration test kit for the most accurate results. It's hard to tell the difference between 25ppm and 50ppm, or 50ppm and 100pm on a test strip.
Scale: