They don't have to rely on Twitter. Politicians have office budgets which include money for sending out mailings to their constituents. Most politicians also have their own email lists to which they send out updates and run polls. There are plenty of venues outside of Twitter to contact constituents.

Twitter itself is under no more of an obligation to publish material from a politician than a newspaper is. Both are corporations that operate under obligations to their shareholders. If a politician really wants to publish something on Twitter, they can pay for promoted ads just like they can with a newspaper. Expecting social media platforms to promote their message for free is hopelessly naive and does not match how democracy has worked for centuries. Sure, newspapers have published political messages for free via editorials and stories, but they are not obligated to do so. Getting the message out has always required time from volunteers and/or money towards supporting the cause.