The dishwasher part is only true for aluminium pots. Stainless is fine in the dishwasher.
The old idea of a patina in the teapot being necessary for a good cup of tea may have some merit, although I have never been able to verify it. Coffee equipment needs to be spotlessly clean and oil free.
If you are using an aluminium pot it may be somewhat difficult to remove the limescale. Maybe someone knows how it's done. The normal descalers are not for use with Al.
A Brita filter does not make distilled or deionised water. If it hasn't been used too long it does soften the water a bit and is supposed to remove bad tastes. A moka pot tends to build up limescale readilly because the relatively small amount water is boiled down to only a small amount left and is usually then left to cool down which gives scale a perfect chance to build up. The scale is harmless in itself. More important is the filter basket and upper part of the pot with the rubber ring and filter disk that should be well cleaned with detergent to remove coffee oils that quickly become rancid. There is no other reasonable way to remove the oils except to use detergent (or for stainless steel the dishwasher or espresso machine cleaner, also a strong detergent). It goes without saying that all parts need to be well rinsed with clear, hot water before using.
Like with everything else, however, you can test this for yourself. Try not cleaning for awhile and see if the taste changes. In my opinion there is no place for dogma in coffee making or in most other areas of life.