Hi @jchaomignano,
While I do not use Google Assignments LTI, I believe that what you have described is accurate.
Why? This is because the Canvas LMS and the Google "ecosystem" have the ability to "talk" with each other but when you take an Canvas assignment that is configured to use Google Assignments LTI in one course and use it again in another Canvas course, the two systems do not "automatically" talk with each other and the Google "ecosystem" does not know that the new instance of the assignment exists.
This also happens with Canvas quizzes that are configured to require Respondus LockDown Browser. When the Canvas quiz is copied to another Canvas course, the Respondus system needs to be told that the new instance of the quiz exists. While not as involved as Canvas assignments using Google Assignments LTI, teachers tell the Respondus system that the new instance of the quiz exists by simply clicking on the "LockDown Browser" link that appears in the Canvas course.
While not an exact answer, I hope this helps to explain why those steps are (probably) necessary.
-Doug