Chemical plastic recycling is viable and creates a product as good as plastic newly created from petroleum. Mechanical, not so much, it leads to poor quality plastic that tears very easily and is unsuitable for food packaging for example due to contamination concerns.
There are two ways chemical recycling can be done:
Burn plastic into carbon dioxide (and water). Capture carbon dioxide. Capture waste heat, distributing it in a district heating network. Create hydrogen from water using electrolysis. Combine hydrogen and carbon dioxide into methane. Post-process the methane to create longer chain hydrocarbons, and eventually the plastic molecules you want.
Create petrochemicals directly from plastic, avoiding the need to convert it to carbon dioxide by burning and later back to hydrocarbons. Those petrochemicals can then be processed to create the needed plastic molecules. At least Alterra Energy has such a process.
Of these options, (2) has probably greater energy efficiency (leading to cheaper recycling), but (1) isn't bad either assuming you have a use for the waste heat. A benefit of (1) is that it allows creating plastic from any waste containing carbon -- cardboard packaging, newspapers, wood and plastic.
So if you want to create plastic from wood or paper waste, (1) is the better option. And if you don't want to recycle plastic to a separate stream (which (2) requires) but rather process a single energy waste stream, (1) is better since it can process nearly any waste apart from glass and metal that don't burn.
Currently most plastic recycling is mechanical, and that explains why recycled plastic has such a crap quality.
I think that given the current processing of plastic waste, it doesn't make sense to separate plastic waste to a separate stream. You are then supporting the creation of inferior quality product, leading to the proliferation of bad quality plastic bags that tear on the first use, leading to people buying a single new plastic bad and using it exactly once, as opposed to using the same plastic bag 100 times (which used to be possible before the disastrous recycling of plastic and usage of the poor recycled plastics in plastic bags began).
But in the future, if technology (2) proves superior to technology (1), then it may begin to make sense to have a separate plastic waste stream.
Also, I'm sure that in the future, new non-recycled plastic will be created not from petroleum but from trees.