Maybe a failing alternator has caused that green gloop to build up on the battery terminals, and eaten away at them, due to the acid build up, but the alternator remains faulty?
Or the continual connect/disconnect of the battery has damaged the alternator?
Or a 10 year old battery is faulty, the terminals of the battery are worn, and rather than mess about fitting new connectors, a new battery would have fitted what was there already? And the alternator still can't charge the broken/faulty battery, because it is still faulty, but more securely clamped?