I presume one could, with present technology, incinerate a stream of mixed plastics, collect the CO2, and turn it back into hydrocarbons. I’m not sure this counts as recycling.

"Not recyclable" in the literal sense. You have lower yields when you reform plastics and there's always contaminants which are difficult (read "we don't know how to") remove. This is a complicated technical challenge. In fact, the recent John Oliver episode was on this topic[0]. There's kinda this weird conspiracy about this, which is more about shifting responsibility.

Though the more honest answer is closer to "mostly not recyclable, a bit not economically viable". But that also depends what we mean by "economically viable". If we're using it in the typical sense of "slightly more expensive for producers" then a small percentage increase. But if we're talking about "if producers had infinite resources" then well still no, but we'd be able to recycle a bit more (we're still not talking much more). I'm assuming the former because the latter is an absurd position.

[0] https://www.youtube.com/watch?v=Fiu9GSOmt8E

What do you mean by recycled in this case? From my understanding you can pretty much only grind it down to garbage filler and maybe press it into some primitive shape. It’ll never become cling film or whatever again

Actually some of the hazardous waste at Hanford is from the PUREX "recycling" process to extract Plutonium for weapons research. A lot of people just don't want to admit that nuclear is very dirty, and while we're potentially able to better capture its waste compared to other waste producing energy sources, our track record with nuclear material in general is very poor.