No, because it’s an email provider and as someone who sends email, I know that if someone posts something spammy or offensive on their platform, I know it could negatively impact other customers, because email is based on an open, reputation-driven protocol.
In other words, someone posting dangerous content could threaten my deliverability, even though I didn’t do anything. Therefore, if I purchase services from an email provider, I will want them to moderate it.
You can make the same argument in favor of allowing racial segregation. In fact, those exact arguments were made. We decided that having an open society was worth it.
You also solve the coordination problem by making all businesses accept minority customers. Southern racists couldn’t boycott all businesses. Likewise, left wing activists can’t boycott you for allowing conservative speech if all of your competitors also have to allow it.
Conservatives currently dominate talk radio, cable TV news ratings, Facebook’s top 10 list of most read articles every week, the White House, and an entire chamber of Congress.
Stop comparing conservative speech to people who were actually oppressed because a large number of people want to use services that have someone moderating the content. It’s offensive.
Vocal conservatives get fired from their jobs in any well paying line of work. Conservative leaning stories are censored from all news outlets except those listened to exclusively by conservatives.
Therefore you wouldn't see any problem with regulations or laws that restrict the rights of companies to enact this sort of policy that Mailchimp is.