Is it possible to use Adobe DTM (aka Satellite Tag Manager) to set basic Omniture/Adobe Analytics variables, like pageName, so we don't need to place this code on the page?
You will need to load the "Analytics" tool to DTM to begin setting up Adobe Analytics variables. The nice thing about DTM is you do not need to migrate all of your analytics code to DTM all at once, you can migrate in steps as long as you tell DTM analytics code already exists on the page.
There are great videos here (under Dynamic Tag Management) which walk you through the setup of DTM and the configuration of the Analytics tool. https://outv.omniture.com/