Can the DTM embed code be placed on the site in another way? Can the DTM footer code be left out?

The DTM embed code has to be implemented in the manner indicated in the documentation. No other implementation type is supported by Adobe. Manipulating the embed code implementation method could disrupt how the engine is designed to function.

For more information on how to implement the code reference the documentation here: https://marketing.adobe.com/resources/help/en_US/dtm/code.html