Not all Custom Fields are available for Issue Collectors. There is some logic to certain field types, such as:
These fields contain information that an unauthenticated user might not be able to see. Other type fields are not available to Issue Collectors:
However, whenever an Administrator sets up a custom Issue Collector, they choose what fields are available to the Issue Collector. Additionally, sometimes Issue Collectors are being used by authenticated users, as occasionally Issue Collectors are set up in the Announcement Banner for use within the JIRA environment. As JIRA Administrators, they should have carte blanche to decide what fields are in their Issue Collector, and what fields are not, and in turn have all fields available for addition to this tool.