The intelligentsia of the 16th century would have been deriding the masses chowing down on oranges while shouting out their favourite lines of Romeo and Juliet and saying that it was populist drivel compared to the Roman/Greek classics
Given that a lot of WS plays deal with issues over royal succession and examine them in some depth at a time when who was going to succeed Elizabeth I was becoming more and more of an issue, I doubt if the intelligentsia would have been ignoring what the ordinary people were going to watch and would be talking about and relating to a current political situation in their country.
Elizabethan theatre was popular with all classes.
It is certainly true that one of his plays, ‘Richard II’, played a part in the Essex rebellion of 1601.
On Saturday 7th February 1601, when the aged Queen Elizabeth was just two years from her death, Shakespeare’s company was asked to perform the play ‘Richard II’ at the Globe Theatre.
The play tells the story of the last two years of Richard II’s reign and how he was deposed by Henry IV, imprisoned and murdered. Shakespeare wrote and published ‘Richard II’ around 1595 but the first editions of the play were printed without an important scene: the Parliament scene or ‘abdication episode’ which shows Richard II resigning his throne. Historically correct, at the time it was considered politically unwise to include the scene because of parallels between the ageing queen and the former king. King Richard had relied heavily on politically powerful favourites, as did Elizabeth; her advisors included Lord Burleigh and his son, Robert Cecil. Also, neither monarch had produced an heir to ensure the succession.
https://www.historic-uk.com/HistoryUK/HistoryofEngland/Shakespeare-Richard-II-Rebellion/