I suspect the issue is that documents created using the Google Apps such as Google Docs exist in the cloud, they don't exist as files that can be downloaded or 'stored' in other platforms such as Canvas' Files. They can only be linked to, such as the assignment you mentioned. This cloud-type document allows for online synchronous collaboration and sharing.
If you want to place a Google Doc in your Canvas Files, you would first have to either convert it to a file such as a .docx file or .txt file. Or you could download it as such and then upload it.
The reason Word docs come through is that a Word doc in Google Drive is just that, a .docx file that is simply stored in Google Drive, it's not a .gdoc file.
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