In the documentation for "Setting up the Mail Server" it mentions that

Hi, the documentation says that conf.suite.ANONYMOUS_ACCESS is deprecated. But at the same time, the same documentation even says that setting this property to true is an pre-requisite for Appian in order to receive emails. So proceed with setting the above property as of now(I believe even there is no other alternative to this), Might be this setting could be changed in future versions.

@Jordan, the setting is deprecated since the application used to allow anonymous access to the platform which is now not available. However, it is a prerequisite for the incoming email access. You also have to enable "Public Events" in the process model you are trying to kick off via email message for the whole process to work.