If you want to regulate large technology companies, just say that. You'll find that plenty of people agree with you. Trying to contort private expression into state expression isn't going to get you very far, either in terms of convincing others or legal action.
(FWIW: conflating "progressive" and "left" belies a fundamental misunderstanding of left-of-center politics. It's a misunderstanding that you're going to want to correct if you want people to entertain your positions.)
Conversely, even without me digging up any references of the progressive politicians also using the power of government to suppress free speech, it seems like you're making a false distinction.
Progressives are using their power to try to get people fired and unemployable (which is somewhat disingenuous to call "social shame"). When you can make someone unemployable in most of an industry, there's little difference between that and using the government directly. Furthermore, progressives use large tech companies (with power comparable to the state in some respects) to censor speech on their de-facto-town-square social media platforms.