The speech in question is prompted by Achilles sending his best friend Patroclus to ask of Nestor how things are going. Nester responds to Patroclus in Book 11: 656-803 of the Iliad. The speech begins with justifiable comments about Achilles indifferences to the suffering of his fellow Achaeans, then recounts Nestor’s greatest exploits as a young man and then ends with recommendations on what Achilles could do beside sulking in his tent.
He is renown among the Achaeans for his eloquence and wise counsel. “Nestor, sweet of speech, the clear-voiced orator of the Pylians, from whose tongue flowed speech sweeter than honey. Two generations of mortal men had passed away in his lifetime, who had been born and reared with him before in sacred Pylos, and he was king among the third." (Iliad 1.247) He might not be strong enough to heft and throw a spear or to pull back a bow string, but he can provide vital strategic advice, lead by example, marshal his troops and vitally at this point in the story, drive a chariot.
First in his speech, Nestor complains that mighty Diomedes took a sharp arrow in the foot, Odysseus received a wound in the side by a spear thrust, Agamemnon took a spear thrust to the lower arm, and Eurypylus too was struck by an arrow in the thigh, “and this man beside (Machaon) have I but now borne forth from the war smitten with an arrow from the string. Yet Achilles, valiant though he be, careth not for the Danaans.” (Iliad 1.660. “ Danaans“ is another poetic term for the Greeks.) Oddly he does not mention warriors injured earlier in the epic like Menelaus after the botched duel with Alexander. ( Book 4.104) Nestor only references those he personally could not aid this particular day. This is an elderly man complaining that Achilles did not do what Nestor could not do himself.
Next, comes a lengthy tale about a war in Pylos during Nestor’s youth. The story in no way compares to situation the Danaans find themselves in currently, nor compares to Achilles current situation. I would follow ”The Paradigmatic Nature of Nestor's Speech in Iliad 11” by Victoria Pedrick in suggesting that this is Nestor’s aristeia in the Iliad. An aristeia is a warriors finest moment in battle. This is the moment when a hero is most akin to the gods. This is his moment of kleos (glory) and if it happens in the Iliad, a chance at unending fame. If Protesilaus (Iliad 2.695) and Ipidamas (Iliad 11.221) can both come to Troy in order to attain kleos in the Iliad, why not Nestor? As Pedrick points out this is the moment in the battle for the ships, where Nestor’s great aristeia affects Patroclus’ return to battle and the course of the war forever. From the moment of Nestor’s aristeia, the tide of battle turns, the Trojans will soon be routed and Hector’s fate sealed.
And finally Nestor suggests what Achilles should do or should let Patroclus do. It sounds a lot like what Nestor wishes he could do. In tone it recalls (Iliad 1.255) when Nestor whined that men better than those of the current age use to listen to his advice.
In summary I would suggest that, like Nestor’s famous speech about chariot-racing in Book 23 which has nothing to do with chariot-racing; Nestor’s famous speech about Achilles in Book 11 has nothing to do with Achilles and everything to do with Nestor.