Creating an entirely new manufacturing organization including top nominees from each major current manufacturer could work together to come up with the design (along with new young consumers for a fresh perspective on laundry washing aesthetic) together to create a line of washers and dryers donated by them to donate for the investment in their future. The recycling company can be in the same location as the sorting and manufacturing facilities so they all have access to the materials. I think I recall learning of one such place in video for an edX Circular Economics course. I'll have to remember to find the link later. (edit: see bottom for link)
Each machine should definitely come with one of these: https://www.greenbiz.com/article/its-ball-keep-microfibers-out-ocean
Another funding resource: Buy the machines from people currently owning them but wanting to sell them so they can lease instead of own. Definitely make sure this washer is top of the line and of the highest quality otherwise people won't want to participate (I would think).
EDIT: From edX: "Coolrec extracts cast iron parts from Miele washing machines and delivers these back to Miele for reuse in their new washing machines. Miele is a German producer of household appliances." https://youtu.be/dz9g-H9NJcs