Ah. I've just been unlucky.
My 2 builds were both old Tappan units. Naturally, I took things apart and had a look. The thermostats on these units were completely enclosed, metal boxes. Sensor into the box, control shaft into the box, contact terminals coming out. No access to the mechanism. No screws, etc.
I considered attempting to pry the thing apart. Then, I figured getting some tab bent properly, or spacer added, or whatever and then finding the little hook that sits on the pivot at the end of the spring (which is now somewhere in a 10 foot radius on the garage floor) and then getting it back together was just over my tedium/budget threshold.
Thanks for the heads up. Next time I'll try to find a unit that allows for some adjustment.