Dang right, doctor! I'm feeling better about this all the time. After the new starter wouldn't start the car until I used jumper cables, I was kicking myself for not trying that in the first place. But now, it seems that if I had tried jumper cables, I almost certainly would have replaced the battery and left the starter in the car. Under that scenario, I might have had a catastrophic starter failure later, such as the situation NCWayne described below.
Altogether, it wasn't particularly expensive and it didn't take too much of my time (for instance, the NAPA parts store was right across the road from the community college where my wife teaches, and since we were down to one vehicle, I was going there morning and evening, anyway).