Genetically engineered recycling bacteria. Many varieties.
Plus (perhaps) some new acids to help them along with harder breaking.
The "natural" means of recycling for the Earth is biological decomposition, over time, breaking down compounds for consumption to create animal and plant bodies, which themselves can be recycled as feed for animals.
I have already read several articles in New Scientist, Scientific American and Science News (all magazines dedicated to pure actual science) about bioremediation, bacteria genetically engineered to to digest (break down) crude oil, concentrate radioactive elements from soil, plants that absorb and concentrate salts from soil in their roots or leaves, bacteria that can break down plastics and in turn become food for something else.
Our own gut bacteria (in basically all animals) is secreted by parietal cells in the stomach, for an adult human the product is two liters of hydrochloric acid a day to break down foods. but notice it is secreted by cells, which we can take as proof that engineered bacteria can secrete strong acids in a slurry tank, that do not destroy themselves but can break down "garbage".
For a fictional extension of this, presume detailed chemistry and understanding of how to genetically engineer bacteria to produce very specific compounds (like acids and catalysts) advance beyond their current capabilities. (Most of our genetic engineering at this point is learning to cut and paste a gene that exists in one organism, into another organism. We are not at the point of saying "I want sulphuric acid" and being able to compute, de novo, DNA sequence[s] that will produce sulphuric acid.)
Also, the fictional extension would have to include the idea that there is a reasonably finite number of engineered bacteria that can always get all of the job done, for recycling anything we produce, perhaps in the hundreds or thousands of different bacteria. That is fictional because we would need "generalizers", like our own parietal cells producing hydrochloric acid, that will break down large chunks of the millions of compounds we must break down. We don't want millions of bacteria for those millions of things.
Now arrange these into a Tree diagram, some hierarchy of treatments in your recycling plant. Each treatment is followed by a sorting of the products produced, different branches of the sorted products undergo more specific treatments, which again are followed by sortings, and more specific treatment.
In our own bodies, the stomach and its strong acid bath and bacteria there dissolve many bonds and turn the food into a slurry. That is then passed to the intestine, where up to 1000 different species of bacteria go to work processing it, so we can extract nutrients from the slurry. (I am being hand-wavy for brevity, see Gut Flora for a more detailed discussion.)
So imagine your recycling plant as an engineered digestive system. Imagine that processing tree as being far more complex, to deal with any garbage humans can deliver, including paint and lacquer and used petroleum products, glass and metal, unused (and used) medicines, industrial acids, everything.
Don't imagine ONE magic bullet. I would imagine this plant as a strong acid bath to start that "cleans" incoming garbage, a mechanical chopper that reduce everything to sand-sized particles, more baths, slurries, magnetic fields, etc that sort and process, sort and process, until you are left with things that DO require heat processes (metals and glass at least), and components, concentrated by those thousands of species of bacteria (most of which work in combinations, not alone), that can be turned back into products, fertilizers, ingredients, fuel (e.g. methane) or (through controlled indoor farming) new plant life (to absorb any carbon emitted in the processes).
Note also that heat and cold can be very effective in breaking down products without producing any carbon, both mechanically and chemically. The summer/winter cycle on Earth erodes mountains; water freezes and expands, and can break rocks doing it. You will notice much food with high moisture content (like fruits) become mushy if (at normal below freezing temps) frozen solid and then thawed; their cells were ruptured. "Freezer burn" is a similar process. Freeze and thaw meat several times, and without any bacterial rotting, it becomes very unappetizing. The same thing goes for heat: Cooking food breaks it down (and more digestible and doubles or triples the available calories, which means easier to digest by gut bacteria) without burning it.
The recycling plant is a giant digestive process. The majority of this is done just as nature does it, but we don't want to wait centuries for nature to get the job done. We do have some mineral products nature won't break down for many centuries; e.g. stainless steel, glass, mineral crystals (like diamond and sapphire we use). Those should be easy enough to identify and process separately on their own branches.
I would say feel free to add some sophisticated AI (not conscious or self-aware, but chemically all-knowing) to aid the sorting, it could use various testing mechanisms (lasers, sonograms, chemical probes) to decide how to break down some of those sand-sized grains.
Don't think that is too much work: Frito-Lay industries has AI devices that literally examine every single potato chip for discolorations before it goes into a bag, they can make these decisions in micro-seconds and process tens of thousands of chips per second. Your recycling AI can automatically investigate any grain it finds resistant to the mechanical processes, acids or other chemical decomposition processes, or unidentifiable and requiring greater attention, even if there are trillions of those grains per day to focus upon.
Plus it is fiction! The point is that this route is plausible, if only because all of our trash is decomposable some way or another in nature. Special circumstances might fossilize or preserve some things, in sap or bogs or ice or whatever, but your recycling plant is more like the typical course of refuse, or really the worst-possible-case course that causes it to be broken down and reabsorbed as soon as possible in nature. Then amped up by genetically engineered bacteria producing whatever we need to break down the most resistant compounds man has developed.