Nick Spencer has summed up the situation exactly.
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Meade telegraphed Lincoln declaring that he had successfully driven Lee out of Union territory. Lincoln's private response bordered on despair.
The Army of the Potomac's command had been brainwashed by McClellan, Burnside & Hooker into believing that Lee would prevail no matter what. In their mindset sending Lee back to Virginia to fight another day was victory.
In July 4th atop Monteagle Mountain & Vicksburg generals who had infused their armies with confidence that they would prevail no matter what were victorious.
Under Grant in 1864 it took the same army 52 days to pin Lee down in Virginia & by December destroy the Army of Tennessee & take Atlanta & March to March the Sea & capture Savannah.
On an 8 1/2" X 11" map of the Western Theater the entire operations of the Armies of the Potomac & Northern Virginia is literally the size of a postage stamp. That is a graphic example of the war Grant fought.
To directly answer the question posed, Meade did not destroy the Army of Northern Virginia was that he had been trained how not to. Famously at the Wilderness Grant observed Meade's HQ whittling. It was not until a panicked colonel arrived shouting that that Lee had struck again.
Grant almost never lost his temper… in a steely eyed I am wearing four stars appearance Grant said it was time that that stopped worry about what Lee was doing & start focusing on what they were going to Lee.
That is how Meade could have destroyed Lee's army.