Well, the PBS article, first of all, is extremely poorly written. How can you write that plastic is "indestructible" and that it "breaks down" into microplastics that enter our bodies, in the same sentence, and not spot the contradiction.
Also, it actually says "All used plastic can be turned into new things, but picking it up, sorting it out and melting it down is expensive." Later down in the article: "Recycling plastic is 'costly,' it [the report] says, and sorting it, the report concludes, is 'infeasible.'"
So, it does sound to me like there's an important distinction here where recycling plastics (or, at least, disposing of it responsibly even when that's costly) is possible, it's just not happening.
Now, my own perspective: I live in Austria and Germany, where governments have been doing collections of plastic trash separately from residual household waste since the 90s, and are asking households to pay steep prices for disposing of residual waste, but not for disposing of plastic. When I lived in Bavaria, disposal of plastics worked through recycling centers, where people had to separate their plastic waste into half a dozen containers for different kinds of plastics under the watchful eyes of employees educating consumers on what goes where, and doing that kind of separation was the only way you could dispose of plastic at all, without paying the very steep price (effectively a fine), for disposing of it through residual household waste. -- There are a lot of recycling centers in Bavaria, usually no more than 10 minutes' drive from where you live, even if you live in a rural area. And they are packed on a Saturday. Nobody can tell me that people aren't using them.
I'm having a hard time imagining that the government would ask households to go to all that trouble and then just send it to landfill. They'd have to make sure to keep it a secret, because people would rightly be angry if they knew this, and they'd have to pay for it through back channels of sorts. This kind of scandal, while it seems possible, hasn't actually broken yet, to the best of my knowledge, over the course of the several decades that this practice has been ongoing an in any of the governments that have been doing this, which are many.
So, my working assumption until this particular scandal actually breaks, is that alarmism coming from environmentalists in the U.S. is really about poor waste-disposal habits on the part of the American consumer, and not some kind of conspiracy by big oil that governments are in on, to make us believe that plastic is anything other than the devil.