This isn't really a question - since I found a solution. It's an observation in case others encounter a similar problem.
We purchased a new ST-LINK/V2 adapter recently and it shipped with firmware "V2.J28.S6".
We use this adapter along with a cable from Tag-Connect to program our boards (STM32F030C8). However, I found that I could only program our board if the cable was connected before I start the Program+Verify operation. (Related problem: the "Automatic mode" does not recognize when the cable is connected to our board - and the utility gets sluggish when I do).
Since the latest ST-LINK Utility included firmware version 27, I decided to "upgrade" my adapter from 28 to 27.
Once I did this, everything worked as expected.
It would appear that ST-LINK/V2 firmware V2.J28.S6 contains some kind of bug that does not exist in V2.J27.S6.
Anyone else seen this?