Perplexing Plastics (De-sorting Garbage)

Not too long ago, someone let me in on a disappointing secret: our plastics are not recycled, they're burned.  I haven't found more details about paper, but based on the sorting in Nishinomiya, a different municipal district from where I live, I would guess the same is true.  It is a little perplexing to me.  Why?  Why do we sort these from the "burnables" if they are burnable?  I would have guessed that they have a separate burning process, however, if they are dirty (for example, a pizza box, or the wrapper from butter) they are put in the regular burnables.  So the regular burnables are able to handle them.  So then they are burnable.  I just don't really understand how this works.  Or why it works.  Apparently Japan only recycles about 21% of it's waste.  PET bottles, cans, and glass.  Why all this sorting???  Should I continue to put in the effort of cleaning my plastics to make them plastic worthy?  Does it matter?  I live in a world where I must abide by rules that don't make sense to me.  I need an existential director to give me my motivation.  Why do I put the plastic container in the plastics?