The OSA is mostly supported by people who read The Guardian or The Times and watch the BBC. It was originally the work of academics (not big tabloid readers usually) like Lorna Woods, who is supposedly a professor of "internet law", a guy who is the founder of Ofcom, and Baroness Beeban Kidron. If you search Google News for their names you will find lots of left leaning broadsheets and not tabloids.
https://news.google.com/search?q=Beeban%20Kidron&hl=en-GB&gl...
The first five news sites for that search are: Financial Times, The Guardian, BBC, The Times, The Guardian again. Zero tabloids.