Far from being mowed down by longbowmen (although they absolutely would have taken a toll, especially on horses), the French cavalry was turned back by the stakes the archers planted in their positions. The stakes were likely mingled with the archers in a kind of thicket, rather than the fence you see in movies or video games, preventing the french knights from seeing the danger until the archers fell back and it was too late.
The French second wave, armored foot, was essentially invulnerable to arrows. But the French piled into dense columns to attack the English men-at-arms (they had no interest in fighting peasant archers, there’s no glory or loot in it). So the front of the column is in big trouble and is being pushed into the English spears by the men behind them, and those in the rear have no idea what’s happening up front so they keep pushing forward. This is how people die at rock concerts or on black friday.
Mostly out of ammo at this point, the English archers come out to gang up on stragglers at the edges of the columns. This is the “flank attack” that eye witnesses report. The peasants certainly didn’t charge the flanks of the columns en masse, they would have been slaughtered. But they could definitely pick off those who tried to escape the press out the sides. Without an escape route, the French columns turn into a panicked rout. This is where most of the French losses happen, either stampeded by their own or cut down or captured by the pursuing English knights.
The English don’t chase them far, however, because there’s still a second line of French men-at-arms which hasn’t been committed to the battle, and the English have a large number of captives to keep an eye on. In the event, the French don’t renew the attack, and send out their heralds to parley.
So yes, the English won a major victory and the peasants played a big role, but the popular version misses the actual mechanics of the battle. “Conquest of France” is also rather strong. Immediately after the battle Henry turned around and went home to London. His later advantage is more down to the long term consequences of the sheer amount of the French aristocracy killed in the battle rather than any immediate gains.