Exeter was one torpedo hit away from sinking. She was down in the bow ; flooding ; listing and burning from end to end with all main guns destroyed or out of action and limping away at 12 knots. It's unfortunate Langsdorff lost his cool and ran away from the fight, since this doomed him. His best chance was to use the second half of his ammo to run down the Exeter and sink her. This would force the balance of Harwood's force to attack and get battered until they were crippled too. Only at that point could AGS escape before the rest of the RN fleet arrived a couple of days later.
Not at all.
If he used up the rest of his ammo on the Exeter, the AGS would be a sitting duck for the larger RN forces on the way, unable to continue the mission, and trying to limp home without primary armament.
What makes you think Harwood would be "forced to attack"? If the Exeter was finished off Harwood would do exactly the same as he did, shadow the AGS while staying out of range of the larger guns.
He did what was the best choice of options, get away to a neutral port, quickly repair battle damage and then try to escape.
Since AGS had been at sea for months her hull was fouled so she could only make 24 knots....but had enough fuel to run for 16 days at that speed. The British Cruisers could only manage that speed for a few days before they would fall out due to lack of fuel and return to port.
The British cruisers only needed to shadow AGS for 36 hours or so, long enough for Force K (and others) to arrive.
AGS had already made several attempts to engage the Ajax Achillies on the run into Mentivideo, without getting any hits, they could almost certainly shadow the AGS while staying out of range.
The question really is could the AGS evade the Allied warships in the 6 hours or so of darkness, and escape detection by Allied aircraft the next morning.
The Achillies carried a Walrus floatplane, and the Cumberland carried 3, the Ajax had a Seafox floatplane.
Given the choice of running or death, it would be a worthwhile risk.
He obviously felt that escape in AGS's current condition was impossible, so attempting to sortie could likely doom most of the 1,000 man crew
At the end of the day naval clashes are determined by command decision, and the near miss that wounded Langsdorff may have influenced his following actions and cost them the battle. Otherwise they had established a clear tactical victory. But without a well planned 'end game escape clause', AGS was doomed.