The critical assumption here is that it is easy to make plastic from "carbon, nitrogen, and oxygen". While those are the primary atoms in plastics, actual polymerization reactions (to form plastic) require them to have specific reactive functional groups. The most common ones are alkenes (polyethylene, polypropylene, polystyrene, PVC, Teflon), epoxides (epoxy), or amines and carboxyl groups (the 2 monomers of nylon). While some compounds with these functional groups exist in nature, they are rare and polymers based on them do not have the full range of properties that conventional polymers do. There is a lot of active work in this field, but even once these polymers are perfected it will likely require quite a bit of chemistry/engineering knowledge to produce thm safely and without huge quantities of waste. The only practical way to get large quantities of the major synthetic polymers is as a by-product of oil refinement (they are in such low concentration in oil that it would be infeasible to refine oil just for them), and again there is the safety concern of bootstrapping it.
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Hmmm, thanks for bringing up those points and clarifying it for me. Hopefully research will pan out ways to make these functional groups efficiently, but as of now we have yet to see. Maybe in the future we could take advantage of genetic engineering and bacteria to produce the groups we need from the raw components...
well micro-fabricators might enable such processes, not all plastics would be a benefit to create. So perhaps a good first step is determining what things would benefit most from plastics and what plastic properties would be needed. That way we would only spend energy of determining the processes for plastics that would actually be beneficial to society.