That's never been part of any conversation about free speech. (Even the broader non 1st amendment version). And it doesn't make sense. You can always take cash or check. The post office doesn't discriminate.
However, moral imperatives are not things we should legislate as they aren't universal, and "the government cannot regulate me" is a far cry from what some are calling for now, which is " the government must regulate you".
Also, just because something could be private doesn't mean it should be run for privatised gain. I'm pretty sure the UK post office was recently privatised.
But that has no bearing on financial services (like square or visa, which are all contract carriers) or hosting companies (also contract carriers), or other things like that. ISPs are the one exception which might fit the bill of a common carrier.
Plenty of times throughout history, oligopolists and monopolists have been forced to follow common-carriage rules, whether by actual regulation or the threat thereof.
Also, you are conflating the First Amendment, a legal instrument, with the ideas of freedom of speech and open discourse. The First Amendment bars the government from itself censoring people. It does not say that the government cannot require non-discrimination on the part of private companies.