This shows that the rest of the bush is healthy. ???

I have had many cases over the years of just a branch becoming chlorotic looking, especially early on in the growing season. The majority of cases were to light bloom colored (white), rose bushes.

Two possible things may be going on.

A weaker cane having some sort of circulatory problem caused by: hidden winter damage (not in your case), old aged cane not able to move nutrients efficiently causing growth from it to be sickly, mechanical damage, insect damage, etc., just cannot support normal, healthy looking growth.

Secondly, a mineral/pH issue in the soil is present, that needs to be corrected, is causing early signs of chlorotic symptoms. The simplest answer to this possibility is a couple, consecutive liquid feedings of either Iron or Acid Loving Miracle-Gro.

Moses.

Thank you, Moses. This year I gave them Osmocote slow-release back in February— with the idea that I wouldn’t have to do additional feeding for 6 mos. I was planning to repeat the Osmocote in early/mid-July. I really want to believe in the Osmocote Way. ;)

I love Osmocote Plus.

I don't love when some roses do that off color leaf trick. Some never do it. It just looks like old leaves.

So trim them? Some roses grow on old wood.
I really like this rose and want to keep her healthy.

I would trim them off.

I'd look for some mechanical damage, and cut below that.

It's also possible it's just an old cane. Rose canes have finite lifespans, and when they are old, they have to go.