It's not just plastics. Practically every substance we use and rely on to maintain civilisation as we know it has a petrochemical or fossil fuel base.

I understand the not burning fossil fuel thing, but why can't it be seen like another mineral resource?

Because that's still geological carbon entering the overall cycle on the surface. The air and ocean are giant buffers of it. When it's needed it needs to be pulled from there somehow (such as by felling trees or directly extracting CO2). Unfortunately that's not economical when it's legal to tap the giant lakes of it sitting underground.