There's no logical reason why connecting the three steps of the process make sense, unless you start with the premise that you really want to bury post consumer plastic and work back from there.

It would only possibly make sense if recycling was fundamentally a dirty process but since we've got enough zero carbon energy available to suck carbon out of the air we've got enough to recycle. We can then use that energy saved by recycling to suck carbon out of the air.

Unless there's some weird process or catalyst that makes atmospheric CO2 to consumer grade plastic the absolutely cheapest way to extract CO2 it'll always make sense to save energy and redirect that energy to the most effective method of getting CO2 out of the air whatever that is. I have no reason to believe plastic production will be that and it seems unlikely.