11.5V = Dead Cell?

^^^This^^^^ I think we're in agreement.

All I'm pointing out is the 11.5 Volts was measured at location that tells nothing about the battery.
Nor does it really tell us much about the alternator even though the engine was running.
It would only be useful in conjuction with other voltage measurements.

Any diagnoses needs to start with step 1 in the above post.