Factually wrong. Both when paying, and when using something "for free", there's a contract in place (terms & conditions, etc.), and that contract also needs to obey local laws. I don't know if you've ever heard of GDPR, but that's one example of things you are entitled to when using a "privately owned platform for free". Even the USA has laws regarding this, though less strong than GDPR (look it up).
> refuse to ship your package based on an any legal criteria they choose
The question here is what is a "legal criteria". And just because the law is a certain way today, doesn't mean that's the way it should be. It used to be legal to kick people out of restaurants because you didn't like the color of their skin, it's not today. It used to be legal to fire people because they were members of a union or other political group, it's not today. (All depending on your country's laws of course.)
Also, the point of an analogy, is to illustrate a point through a comparable situation. Pointing out that there's a "massive difference" but never explaining what the difference is and how it affects the validity of the comparison, is just a (very common) misunderstanding of what an analogy even is.