While you're correct that cross contamination is a huge issue, it's the lack of proper plastic recycling infrastructure that keeps most localities in the US from recycling types 3-7. Once China stopped importing our trash, it meant that it didn't matter how you sent in your plastic recycling- it was getting sent to the landfill by the recycling center anyway.

Attempting to shame individuals into having to separate and haul their own recycling is also an extremely inefficient and ineffective way to go about improving recycling rates. If you want people to recycle en masse, make it free, convenient, and accessible. If poor people who don't have cars, work multiple jobs, and barely have time to take care of their kids can't do it, then your solution has elements of classism, racism, and sexism, that will render it moot.

Instead, demand that producers, not consumers, take responsibility for their own waste. Producer take back laws effectively place the time, energy, and financial externalities of dealing with waste back onto the producer, as opposed to the time strapped, survival focused, resource limited consumer who should be presented with better choices anyway.