Honestly that whole "cuts down trash" thing misses the point. Washing a reusable filter uses extra water and energy for hot water every single day. The carbon footprint from heating that water probably cancels out any trash savings. Plus those metal filters are shipped from factories using resources too. Sometimes the simple paper filter you toss in compost is the better choice.
But what if the paper filter is the easier choice for most people? The water and energy to wash a metal one adds up, and not everyone can compost. Sometimes the simple answer is just simpler.