In fact most people only read them that way, and in such contexts. The vast majority of students who study Shakespeare do so in courses offered by English, not Drama, departments. But it bears repeating that the literary approach can lead us to forget that the primary aesthetic object is not the script , but the evanescent performance of it. Plays get put on, acted out. Directors make stage-pictures.
Artists make picture pictures.
Playwright is spelled like wheelwright, so designating the play as a made, not a written thing. Narrative pictures are constructed, built, just like wagon wheels and dramas. These are related activities. The director, the playwright, and the narrative artist all seek to make an action manifest. Consider the early Renaissance painter Masaccio.