Someone will come up with something. I'm starting to see Bamboo straws in AZ because CA outlawed plastic straws. The companies that are making these alternative products are having a boon because of the new laws. If we required take out packaging to not include styrofoam or plastic in a couple years I bet there would be some incredible innovations.

This is still a pretty good solution. The amount of plastic used is cut dramatically, obviously, and as long as the containers are made correctly they can still be recycled in a normal paper process.

Is cardboard that is dirtied with food oils actually recyclable? I thought this is not the case.

If it's made correctly, yes. Pizza boxes aren't (because they aren't coated, so the oil soaks in), but e.g. orange juice cartons are.

Orange juice cartons are surely plastic-lined or foil-lined too though

Right, which is why recyclability depends on the way they're made. The PE formulation used for things like juice cartons in the US can be broken down when the cartons are pulped.