For the sake of discussion, would you feel the same way if the CEOs of AT&T, Verizon, and Comcast woke up tomorrow and decided they want to unilaterally block all of their customers from accessing any news sites they deem to be against their political leanings (just for the sake of discussion, let's say they block all access to MSNBC, CNN, WaPo, HuffPo, and the like).
If that happened, it would effectively cut off most Americans from being able to access news unless it agreed with those CEOs' political agenda. Would we all still be saying "well it's totally fine because AT&T et al are private companies"?
It seems obvious to me that sometimes companies become so large and have so much influence that they should not be allowed to just ban/block whoever they want, even if they are private. I'm not sure if AWS in particular is at that point, but the App Store and Google Play Store certainly feel close.
I'm not sure what the answer is, but I don't think this "they're private so they have no obligations to serve you" is valid at the scale of the companies in question.