Note that the policy says you have access to "a" lounge; not necessarily your choice of lounges when there's more than one. As long as KE welcomes you to their lounge when flying them, the policy is satisfied.

I've had similar experience with the AF JFK lounge, being directed to the AZ lounge when flying MU. It wasn't that long before boarding, otherwise I would have trumped their policy with Priority Pass. Be aware though that AF and KE have specific hours for PP cardholders, with some gaps in coverage.