No, the button band looked no better with a fresh pair of eyes. I unpicked the seam, went and watched a better tutorial video, and had another go. (There’s plenty of tutorials on using mattress stitch for vertical seams in knitting, but not many that are helpful when the two edges have different gauge and go in different directions. The one I finally found was great, though.)

The seam is nearly invisible and it’s like the two pieces are one. That’s the way it’s supposed to look like. New version on the left (or top, if you’re on a small screen); old on the right/bottom.