Shakespeare was not primarily a playwright. If nearly your entire body of work is in iambic pentameter, you are not primarily a playwright. He was incidentally a playwright, and he was primarily a poet.
It is not a different game. You write in English. He wrote in English. You both make, therefore, an implicit claim to wish to communicate with people who read what you write. It is not a different game.
If not one in a thousand Americans can read him without a glossary then he should serve the perfectest example to you. What he wrote is, at present, generally insensible, but it is replete with meaning to those with the interest to divine it.