Let's follow that logic: the internet infrastructure is owned by private companies (i.e. Comcast), so shouldn't Comcast be able to censor and throttle anything that goes over their network?
So, in order to prevent that situation, we allow one company a monopoly on the market. In exchange, that company is forced to follow common carrier rules, such as not interfering with speech.
We all agree to use the same company for infrastructure, just like we all agree to use the same couple of companies for social media. It's just when the social media companies do things that are bad people pop up and say "they are private companies in a free market they can do whatever they want" because these companies are new and cable companies are old.