soql – Why is an API User with R/W access to a lookup field, getting an INVALID_FIELD error?

My API user makes the following query in Salesforce via the Standard REST API. They have Read / Edit privileges on the Contact__c field and Modify All on the Channel__c object. They do not have access to the Contact object

SELECT Id, Contact__c FROM Channel__c LIMIT 1

They get this error:

INVALID_FIELD: SELECT Id, Contact__c FROM Channel__c ^ ERROR at Row:1:Column:12 No such column ‘Contact__c’ on entity ‘Channel__c’. If you are attempting to use a custom field, be sure to append the ‘__c’ after the custom field name. Please reference your WSDL or the describe call for the appropriate names.

Any ideas why?