I suppose it stands to reason that if DL can restrict access to third party operated lounges to J pax only, so can the other Skyteam carriers. I do somewhat understand the frustration however as KLM does not document this fact anywhere on their website.
On a side note, I was flying on KE DPS-ICN in coach last month and the KE agent gave myself and spouse lounge invitations for the third-party "Premier" lounge. It seemed a bit odd as Skyteam member Garuda Indonesia has a lounge right next door. Why wouldn't KE just direct me to use the Garuda lounge?
KE doesn't likely want to pay Garuda what they want to charge. Plenty of airlines don't use the lounges of their alliance partners for their contracted lounge. If you went over to Garuda with your DL card they would have let you in. However DL and not KE would have been paying for the admission.