A poorly or improperly designed signal conditioning circuit could fail with a LOW battery. Even thought there is enough voltage to sorta work. Most of us in the field design our circuits to work to 1/2 battery voltage. But not everyone does that. Let me tell you that there are times when it is not that easy to do either. That is the typical design target though. Let's say that you are a newbie design engineer that has been assigned to create an automotive circuit but you don't know about this rule of thumb. Or marketing and your lead engineer are on your case to get it done fast. You might scrimp on the rating of a critical component, choose an inappropriate component, or not add an important component. So this design goes out into the field and occasionally, it's trying to run with a low voltage battery. That causes many of the circuits to have to operate in a linear mode instead of a digital. Linear draws too much current, overheats components, etc. Circuit components, not wanting to be in the linear region, blow.
So, that's how a 'dead' (but usually just a low) battery can cause failures.
BTW, when I'm designing circuits for automotive use, I also have to consider 'Load dumps' which cause 60V spikes to occur on the battery circuitry. That's actually easier to handle then dealing with low battery voltage. Especially chronic low voltage.