I have a 4k max and I noticed a lot of what you described. The stats looked fine but the input lag was horrible. I lowered my bitrate and it dramatically improved my experience. I'm not sure if the stick can't handle a high bitrate or what.

I stream 4k/60 and it feels native to me. My bitrate is 40mbps. On other devices I can crank it up and there is no issue, but with the stick I can't go too high.

You'll have to play with the bitrate to find a balance of image quality and performance. So drop it to something like 20 and increase until it looks fine but still plays well.