Earlier quoted context omitted.
If you wash up manually in a bowl, you can use the remnant liquid to do your recyclable washing before pouring it down the sink, but .. well, as you suggest "I ain't got no time for that". As for your pizza box idea, I'm not sure that's not simply complicating things - now you'd have two independently-recyclable materials tha most consumers will simply leave together and dump in the box, leaving you with sorting prob…
I've heard that Pringles tubes (foil lined cardboard with plastic lids) are a bit of a recycling nightmare for that reason.
Drives me nuts!