I'm not sad about my plan taking a new direction. The fabric these shirts is made out of has stretched unevenly or pilled or otherwise made these good nice-enough-to-wear-out-but-I-don't-mind-if-I-stain-it shirts.
I did experiment some when attaching these sleeves. I discovered that if I run the basting stitch all the way around the sleeve cap, leaving only the underarm seam allowance free from the basting stitch, I can gather the sleeve cap enough to fit it into the arm hole with a minimum of gathering showing. Of course, I risk having a stray gather show on the front or back where gathers are unexpected, but even so, I like this better than having all the ease fullness at the top of the shoulder. Sometimes, it works out perfectly so that the sleeve sets in with no gathers showing. And then I rejoice.