I've never brought a washer/dryer from one house to the next when moving. In my experience, it's been fine. Sometimes I had a washer or dryer that I didn't like all that much (took longer than I'd like, confusing controls, etc.), but I've never ended up with one that damaged my clothes or was so bad that I needed to replace it.
I'm sure there are many instances where there are replacement-worthy problems, and there certainly are people who have splurged on really nice appliances and will want to take them with them when they move, but I think those situations are uncommon enough that, overall, people save time and money by not lugging their washers and dryers with them every time they move. (In places where it's common not to move them, at least.)
Every time I have moved the contract stated all appliances stay. The legal default is you take them all - washer, dryer, stove, refrigerator, dishwasher, light bulbs, window shades. However realtors all know to put into the contract what stays because most most people want those things in the house and don't want to move them. However more than once I've bought a house without something (the old fridge broke a week before they listed the house so they didn't bother replacing it). If something isn't in the contract and you leave it behind they can charge you for disposal of their garbage!
I'm sure there are many instances where there are replacement-worthy problems, and there certainly are people who have splurged on really nice appliances and will want to take them with them when they move, but I think those situations are uncommon enough that, overall, people save time and money by not lugging their washers and dryers with them every time they move. (In places where it's common not to move them, at least.)