Is it possible to produce plastics that don't degrade or return CO2 to the environment? Make everything out of plastic, and then bury it in landfills, seems like one way to capture carbon.
Let's not. Even if you could make this work, sort of, plastics have all sorts of other nasty environmental and biological effects that we should try to avoid - they tend to break into pieces tiny enough to get everywhere in an organism, and then mimic other signaling chemicals well enough to fool systems. There's plenty of writing on microplastics you can dig into if you want to learn more.
Making less out of plastic would be a good step. Making more out of plastic isn't a good idea.
Also just producing less useless plastic. Much packaging can be made out of cardboard. If it needs some water protection, a thin wrapping of plastic is plenty.
The one thing I'm not sure of is food packaging. There's a lot of plastic used in that, but I don't know if we have any alternatives. It's tough to package food because you want much of it to be sealed air-tight. But food packaging does have an advantage because it only has to last as long as the food inside.
I think you're stuck between the anti-plastic and anti-GHG ideologies. This reminds me of when anti-nuclear collided with anti-GHG. Not everything has a single obvious right answer. There are always tradeoffs. In this particular case, the environmental problems are easily solved by using landfills that don't release anything into the environment.
Having said that, plastic production itself is a far bigger GHG emitter than burning it. Plastic product production emits 1.5 Gt of CO2e per year while global CO2 emissions are 35 Gt [1]. That's 4% of global emissions coming from plastic production. Incineration is hardly anything in comparison.
That carbon would have to come from the atmosphere in order to be "capture". Converting oil products (already-captured carbon) into plastics is the wrong direction. It doesn't make sense to take a step back just to attempt a step forward.
Turning atmospheric CO2 into plastic probably isn't as feasible as turning CO2 into some other stable form directly. Diverting petroleum production into plastic and them making sure it ends up in landfills rather than the sea[1] does seem like a good thing to the extent that plastic production funges against fuel production, I'm not sure to what extent these rely on separating different parts of the oil that comes out of the ground in the refinery for different uses rather than having one common pool that could be used for either. But plastic buried in landfills does seem to remain out of the biosphere for long enough to be a useful repository of carbon.
Plastics can be made from starch which would capture CO2. However, bioplastics tend to degrade. Making plastic from oil as the vast majority are doesn't sequester CO2.