For an excellent and well documented (from original sources) account of the retreat, the pursuit, and the conditions faced by both armies, read Wittenberg, Petruzzi and Nugent's enthralling One Continuous Fight: The Retreat from Gettysburg and the Pursuit of Lee's Army of Northern Virginia, July 4-14, 1863. There was a lot going on during those ten days!
I strongly recommend you read this account before judging Meade and the US troops! It's a great read regardless of the conclusions you end up with!
Just a few things to consider--what if Grant had just taken command, completely unexpectedly, five days before and necessarily inherited subordinate staff who, he knew, actively disliked him; he had just lost three corps commanders, two of whom were his most reliable generals; if, after a horrendous three day battle with enormous casualties he was out of everything and had to march out of the direct route to the Potomac crossings to resupply; if the replacement troops sent him were completely green and unreliable militia; what happens to gear (already worn out from forced marches to reach Gettysburg) subjected to more forced marches in days of pouring rain (think what that does to shoes),; the fact that Lee was well entrenched in excellent defensive positions before the US forces arrived at the Potomac. I doubt if Grant would have done much better with the combination of all these factors (and a few more, like an incompetent cavalry commander who didn't even try to coordinate his troops).
Of course, Meade was not Grant. And there was this McClellan-type culture of caution ingrained into the Army of the Potomac. Even Grant couldn't overcome that!