Yet another (positive) data point from earlier this week. I'm 1K, arrived into MUC in UA biz (upgraded with GPU, fwiw), and was granted access to both LH lounges (I had breakfast in the SEN lounge and then got a shower in the business lounge). When I arrived, I showed the agent at the door my ORD-MUC boarding pass and my 1K card. He asked "Do you have a connecting boarding pass?" I said "No, I'm arriving in Munich, and I'd just like to take a quick shower." He typed away for a minute and admitted me to the lounge.
So, this is consistent with the narrative that (1) it's an exception, but (2) LH agents are willing to make it surprisingly often. Although as villox's experience indicates, sometimes you don't get lucky. So maybe this is the ultimate YMMV.