The issue is moreso that they exclusively control their brand, and by letting someone else - who doesn't work for them and need not listen to them - control part of that, they are indirectly loosening control of their brand and narrative. They probably do not want the additional headache of buying this product, onboarding it, managing it, etc. The easiest but unfortunate solution here, to limit risk and liability, is to shut it down.
If this is the case, then they are truly beholden to their own lawyers (and not the other way around, which is just ridiculous).
Why aren't they insisting that FEMA not use their brand when talking about "the waffle house index"?