I keep seeing these sort of statements but have yet to see one backed up by a link to some reputable evidence.
My local supermarket accepts clean soft plastics for recycling and when I investigated I found membership of schemes to ensure full transparency. Looking further I found the companies accepting the waste and financial statements indicating heavy investment in machinery to deal with it.
Plastic recycling, because of what plastics are, is basically impossible at cost. PET recycling might make you think "see, we can recycle plastics" but first and foremost, PET recycling isn't recycling, it's reuse (using the PET as a base material to make something else, like fleece, which cannot be recycled), and second: the majority of plastics aren't PET and literally have no recycling path.
I was hoping for something academic rather than a YouTube video.
And the final message is definitely a little more nuanced than that =D