You're right. But, I work in lithography, and there are only positives for going to rectangular wafers. We will be able to clamp is better, I think acceleration stresses will be easier to manage, modeling will be easier because it will be per field exactly the same, and so on. Litho industry (which basically means ASML at this point) might jump on this.

But I'll reiterate I mean all this on a lighter note, don't think it will happen within my career :).

Edit: Didn't read the question you asked in the first sentence. The left over parts are usually just taken along in the process till the end and are scapped when we get to the cutting stage. The problems arise because the structures on the partial dies are not the same as the full dies in the middle of the wafer. This causes a bunch of weird stresses on the edge and in my small corner of engineering we optimize the fuck out of the edge dies so their stresses are less weird.