I don't buy this argument.
The internet providers are de facto pretty close to the common carrier status (except maybe in cases of huge scale, like Netflix or google, where explicit net neutrality might be desirable). There are no huge technical or financial barriers to entry if you want to build a platform for communication over the internet, nothing comparable to e.g. telecom networks (either in 1950s or now).
The cost of acquiring a millions-strong audience, operating at that scale, and becoming essential for large swaths of society can be pretty low, both in financing and manpower; if Instagram is not a good example for you, take more extreme examples of Wikipedia or SciHub.
Would you like newspapers (e.g. that very NYT) be governmentally regulated because they are "too important for society"? It's dangerously close to the late Soviet Union. How effective NYT would be e.g. in uncovering the Watergate case were they regulated "for the common good"?
There's a particular fallacy that makes people think that some centralized regulation would be more efficient and lead to better outcomes than self-regulation on a level playing field. The efficiency is there in very few cases, mostly those that seriously defy common sense (e.g. control over antibiotics), or set safety guidelines against mass illness (e.g. food safety control). In most other cases, a bureaucratic body has as skewed incentives as a for-profit corporation, but much fewer reasons to improve anything.