Every electronics tech will tell you a similar story: the system isn't working and two components are the likely culprits. Component A and Component B are removed for testing. Component A fails unit test and Component B passes. Changing Component A doesn't fix the problem, so Component B is replaced. It turns out that Component B was the problem even though it passed unit testing. Just because something tests "bad" doesn't mean it is causing your problem.
It's also common in electronics for a problem in one unit to propagate to other units. For example, a shorted diode in one unit can cause a relay failure in another unit. Avionics techs who like to swap out boxes rather than troubleshoot problems often find themselves in big trouble when their box-swapping results in blown units in multiple aircraft as components are swapped from one aircraft to another.
I've never heard of an automotive starter failing from a bad battery, but supposedly a lot of motorcycles are susceptible to this. The theory is that a weak battery will keep the starter drive from fully engaging, with the result that the drive is damaged when the engine starts. Lightweight aircraft starters are also known to be prone to failure from "kickback" during a failed start.
A few years ago I was working on our Cadillac Seville, which hadn't run for over a year. It wouldn't crank at all, so I assumed the battery was bad. Replacing the battery didn't fix the problem, so I changed out the starter. It still wouldn't crank, so I removed the serpentine belt and it fired right up. The A/C compressor and clutch were completely seized.