You're confusing him with King John II, who was the one captured at Poitiers. King Philip VI was never captured.

As for why Edward did not usurp the throne after this, it's complicated, and goes back to the origins of why the war was started. Edward did not have much interest or expectation in actually becoming the King of France. Upon his ascension to the throne of England in 1327 he traveled to France to swear fealty to King Philip VI for his land in Aquitaine, which the King of England held not as separate territory but as a fief of France. This act therefore both legitimized Philip as the rightful King of France from an English perspective, and also confirmed that the English would not make any further attempts to assert their claim to the throne through Isabella (Edward's mother, and the daughter of the previous King of France). When, in 1337, Philip demanded that Edward come to Paris to argue in court as to why he had a rightful claim to Aquitaine, Edward refused to go, believing it was merely a ploy to legitimise the seizure of Aquitaine by claiming they'd given Edward a fair chance to defend himself. Philip promptly sent troops into Aquitaine and occupied most of the towns and cities. Now, were Edward to declare war on the King of France over the issue of ownership of Aquitaine, it would technically be the act of a rebellious vassal fighting his rightful lord, especially since Edward already acknowledged Philip as his rightful lord for Aquitaine a decade earlier.

To get around this awkward legal trapping, Edward decided to dredge up his old claim to the throne of France and present himself as the rightful King of France. Even though the French government in Paris obviously dismissed this as nonsense, and Edward didn't have any legal grounds to do this since he'd dropped the claim years ago, he still insisted on calling himself the King of France, and in Brugges in 1340 he was proclaimed as the King of France by an assortment of Flemish noblemen and bishops. This was done for two reasons: firstly, by claiming to be the rightful King of France, he was now no longer - in some people's eyes - a rebellious vassal, by a claimant to the throne, which was more respected. And secondly, it encouraged several lords in Flanders and Dutch-speaking territories to form alliances with Edward; they felt more comfortable fighting against King Philip in the name of a claimant king rather than a mere rebellious vassal. During these events, there are basically no contemporary first-hand accounts that reveal that Edward genuinely desired to usurp the throne of France; he was being totally realpolitik about it.

Fast-forward 16 years to when King John II is captured, and Edward was more concerned with territorial aggrandizement than anything else. He wanted to restore the borders of the old Angevin Empire, and to put pressure on the French government he repeatedly wielded his claim to the throne as a bargaining chip, promising to give it up as soon as they gave him the territory he wanted. He did this so blatantly and frequently that contemporary French writers were disgusted at how cynical and disingenuous his supposed claim to the throne of France was. When John was captured and sent to England as a prisoner, his son the Dauphin (later King Charles V) was made the official Regent of France and was tasked with ransoming his father and entering into negotiations with England. Never once during these long and complicated negotiations did Edward ever demand to become the sole King of France; his entire list of demands was just territory, and territory to be held outside of the jurisdiction of the French Crown, at that. When, with the Treaty of Bretigne signed in 1361, these territories were ceded to Edward on paper, he was just about to formally give up his claim to the throne of France, but stopped at the last minute because much of his new territory was still garrisoned by French troops who refused to leave. Edward demanded that the government in Paris order these troops to withdraw, but they refused to in most cases, and so he kept his claim open and the war would continue indefinitely while he tried to take these territories by force now.

The first King of England to actually attempt to seize the throne of France itself rather than just use it as a bargaining chip was Henry V, almost forty years after the death of Edward III. Neither of the two kings in between them - Richard II and Henry IV - cared much for this claim, and were content to just leave it gathering dust in the corner. This gives an indication as to the real thought process behind holding onto the claim, at least until the 1410s.

Source: The Hundred Years War, volumes I-II by Jonathan Sumption

Hey there,

Just to let you know, your question is fine, and we're letting it stand. However, you should be aware that questions framed as 'Why didn't X do Y' relatively often don't get an answer that meets our standards (in our experience as moderators). There are a few reasons for this. Firstly, it often can be difficult to prove the counterfactual: historians know much more about what happened than what might have happened. Secondly, 'why didn't X do Y' questions are sometimes phrased in an ahistorical way. It's worth remembering that people in the past couldn't see into the future, and they generally didn't have all the information we now have about their situations; things that look obvious now didn't necessarily look that way at the time.

If you end up not getting a response after a day or two, consider asking a new question focusing instead on why what happened did happen (rather than why what didn't happen didn't happen) - this kind of question is more likely to get a response in our experience. Hope this helps!