Interesting system, and it does seem more fair. I agree with your objections -- I find it really hard to cook with dirty dishes, so oftentimes I wash dishes before cooking. That means the person with dish duty is only washing dishes associated with the meal I made. But if the person cooking for me doesn't wash dishes beforehand, then dish duty can include dishes from a day or more before as well.

My partner and I don't have an explicit system. More often than not if one of us is doing more than their fair share of housework generally, that person will complain and the other person will start picking up their slack. I guess it works? Lately, I've been doing way less housework because of my repetitive strain injuries. I think I could count on two hands the number of dishes I've washed in the last couple of months. It sucks! Well, less so for me, obviously. But I do feel bad about it.