When "partners are using six or more LMSs, and are strongly set on using their specific LMSs," is the issue, it seems. One possible solution to this would be to create six separate instances of each course for the six different LMSs. How many instructional designer hours is actually require to duplicate a course from one LMS to another, on average. Even if their are more than six LMSs involved, I think savings would accrue to the institutions compared to connecting via LTI. That would require each institution or group of institutions using the same LMS, then, to support the use of the course in that LMS. The advantage of connecting to the course via an LTI link is that the faculty and staff of each institution aren't required to do much. If they want analytics about student work in the course, it's available for a fee from the course provider via LTI ( about $25US per student per course.)
What gets lost using the LTI linking method, though, is the ability of the instructor to use all of the increasingly useful and nuanced features of the LMS to teach in the way that is best for their particular circumstances. If the argument is that providing the instructor that much flexibility isn't desirable and maybe in some cases, explicitly prohibited then linking to more static content via LTI may, in fact, be what's best.
What I was hoping for in starting this thread was more articulation of those specific teaching abilities that having the content localized in the LMS provide, and comparing those to the specific advantages that exist for connecting to the content that is in some other platform that is never to be referred to as an LMS. So far, I don't see any specific advantages from a andragogical or pedagogical point of view in using the other platform via LTI. There are certainly some efficiencies in the distribution of the content to the instructors and students, but efficiency of distribution does not generally provide lasting benefit to teachers and students.
I'm curious about what types of graded assignments are available in the 'other' platform. The LMS with which I'm most familiar has a very long list of ways to collect student work and many elegant ways assign a grade to that work.