I'm having this problem as well, it has nothing to do with the campaign. My Titan hasn't completed any of the quests on the Moon, but still has access to flashpoint just for being over 900LL. I've finished it on my Hunter and that one can't even access the Flashpoint and is 911LL
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I've experienced similar things, even spoken with others who haven't touched the new expansion. I've even tried switching platforms- where I don't own Shadowkeep- and I still lack access.