In the following document:

https://resources.docs.salesforce.com/198/latest/en-us/sfdc/pdf/api_rest.pdf
on page 21 it states:

Partners, who wish to get an OAuth consumer Id for authentication, can contact Salesforce

I am reading this as I can make a request to Salesforce for a consumer id. I assume that they will provide me with a consumer id and consumer secret which would allow me to initiate the OAuth flow on behalf of any Salesforce organization without them having to install a connected app.

Is this correct? 

Has anyone been able to make REST calls without requiring the installation of connected app through the marketplace?