I can't speak for everyone, Ric, but I do believe I sold you the pair of K1K you are in the process of coming to terms with maybe getting rid of.
Do it. Get the JH13. They are honestly better headphones. There are truly transcendental moments on the K1Ks when everything seems right with the world and the heavens open, but they're marred by all the other things you spoke of. The fact that some recordings can be so good, so well suited, just makes the other recordings feel less worthy somehow.
The JH fixed that problem, for me. If we could trade headphones and I could get my old K1Ks back, I wouldn't. They're my favorite dynamic ever, but they're not good enough at enough to make them truly indispensable, as the JH13 has become to me. I can only assume the 16+3a is even better.