EDIT: I just saw that the title says Tableau Admin. That changes things up, of course. Did you already try to put https:// in the Tableau Server URL when logging in via Tableau Desktop before trying to publish your data source?
Publishing content on Tableau Server has two types: publishing workbooks and publishing data sources seperately. This is because Tableau Server treats workbooks and data sources as two different content types, and each content type has its own set of permissions. So indeed, you could publish packaged workbooks with data sources in them, and it may work.
More info can be found here: Permissions Reference
To publish on Tableau Server translates to the "Save" permission on the permissions reference list.
(My experience is based on Tableau Server 10.5, I'm not sure whether the permission models and its underlying flexibility has changed much with the current re-design of the Site Roles.)
Thanks for the reply. Although this was not the case up until the i published the content on Tableau 10.5 server.
Also, as ideally my role is Server Administrator, its strange that Tableau doesn't allow publish right separately for the data source. I can publish both workbook and data source together. Although there are many / 90% of instances that require to publish the data source separately.
I edited my post just shortly before you replied - in case you may have missed it, my suggestion for now is to try and use https:// on the Tableau Server URL when logging in via Tableau Desktop.
Based on my own experience as a Tableau Server Admin, this trick has worked on occasions where permissions were denied during scenarios that it would seem illogical.