In very rare instances, yes.
You would be hard pressed to get the Supreme Court to rule that the same standards could be applied to a media company. The fairness doctrine met similar objections, and that was never applied to companies using cable, just public broadcasts. In this particular instance, we're talking about Twitch ending partner relationships. It's a hard sell to say that Net Neutrality or rationing limited airwaves is the same as forcing Twitch to pay a streamer money or to put them in a privileged position above other streamers.
And if your plan is to create the same kind of restrictions that you see around, say, Net Neutrality, then that still isn't going to create the environment that you want, because Net Neutrality also protects stuff like harassment, disturbing/gross content, and pornography -- it isn't narrowly applied to a privileged sub-category of speech. In other words, common carrier laws protect your political speech, but they're not specifically designed to protect your political speech. They also protect a lot of stuff that people don't want to see on their Facebook feed.
Even more broadly though, when we talk about revising the Constitution, we're talking about free speech ideals, not just legislation. And while the legislative conversation is worth having, it's also worth acknowledging that community moderation is a free speech ideal. The entire marketplace of ideas is based on the philosophy that some ideas win and lose, and that some ideas become unpopular over time. The entire point of Freedom of Association is that people can form communities and have productive conversations on diverse topics without getting constantly shouted down or harassed. Community moderation and subgroups and forums with strict rules are part of how we encourage a diversity of opinions online; they're why we can have this conversation on Hackernews without someone jumping in and trying to sell penis enlargement pills or telling us all to kill ourselves.
So on one hand if we're having a conversation about laws, it is going to be very difficult to get the Supreme Court to go along with most of the "neutral platform" proposals I've heard[0]. On the other hand if we're having a conversation about ideals, Freedom of Association is something we should care about in and of itself, and we should want to preserve that as much as possible -- we should only be looking at getting rid of Freedom of Association in places where we literally have no other choice. Compelled speech is extremely dangerous, it shouldn't be the first thing we try.
And luckily, we have a lot of precedent on alternative ways to get speech to flourish online. When people refer back to the "good old days" when the Internet cared about speech, there was plenty of moderation back then. There were just also a bunch of independent IRC rooms, independent websites, more freedom to be pseudononymous, and everyone's entire life wasn't tied to a single Google account that could be shut down at any time. So the first thing we try should be to increase platform diversity, because we already have compelling evidence that platform diversity increases speech.
[0]: Particularly a Conservative Supreme Court with people like Barrett, who are probably not inclined to say that 1st Amendment protections shouldn't apply to companies that directly interact with consumers. With the exception of people like Thomas, I do not see much eagerness from Conservative justices to strip free speech protections from companies.