I'll confirm a couple of points first.
On June 28, 1863, Lee's three corps commanders were Ewell, Hill and Longstreet.
Ewell, as you said, was north of Gettysburg, near Carlisle and York.
Hill was on the road which is today US 30, just west of South Mountain about 20 miles west of Gettysburg.
Longstreet was in Carlisle, with Lee, about five miles behind Hill.
First contact was about 7:30 AM, July 1, (DAY 1) between lead units of Hill's forces and Union cavalry, about 3 miles west of Gettysburg.
At 9:30 AM, Hill's first major assault, now two miles west of Gettysburg.
Ewell arrived from the north a bit later, his first units assaults beginning around 1:30 PM (still DAY 1).
Longstreet's units began arriving the next morning, July 2, (DAY 2) but his attacks did not begin until after 4:00 PM.
Longstreet's problem was not so much distance, as the fact that the only road passes through a narrow gap in South Mountain, through which both Hill's & Longstreet's corps must pass.
I have driven that road (US 30), and tried to imagine 50,000 troops with artillery and supply wagons filing through it.
Even on today's road, it would take time.
Why did Lee march north in June 1863?
Bowden and Ward argue at length that this was the South's "Last Chance for Victory," and they define "victory" as another major defeat of Union forces which would force the North to accept a political compromise.
And we must first remember that before Gettysburg was Chancellorsville, in early May 1863 -- in which Lee handily defeated a Union force over twice his size (130,000 versus Lee's 60,000 on the battlefield).
So both Lee and his troops thought they were on a roll, and could win any battle they fought.
A short explanation comes from Lee's after action report:
"Upon the retreat of the Federal Army commanded by Major General Hooker from Chancellorsville, it reoccupied the ground north of the Rappahannock opposite Fredericksburg, where it could not be attacked except at a disadvantage. It was determined to draw it from this position, and if practicable to transfer the scene of hostilities beyond the Potomac..."
So Lee was trying to force the Union Army onto ground that better suited Lee.
As to why Lee lost the Battle of Gettysburg, B&W argue Lee could well have won it, if the battle had been fought as Lee intended.
They provide a long list of failures, nearly all of which were ultimately Lee's responsibility.
But the fact remains that each of Lee's corps commanders in turn, plus Stuart of course, let Lee down at critical points in the battle.
Why they did so has been a matter on endless debate, these now nearly 145 years.
I'll give the final words to Lee. June 8, 1863, in a letter to Secretary of War Seddon:
"As far as I can judge, there is nothing to be gained by this army remaining quietly on the defensive, which it must do unless it can be re-enforced. I am aware that there is difficulty and hazard in taking the aggressive with so large an army in its front, intrenched behind a river, where it cannot be advantageously attacked.
"Unless it can be drawn out in a position to be assailed, it will take its own time to prepare and strengthen itself to renew its advance upon Richmond, and force this army back within the intrenchments of that city.
"This may be the result in any event; still, I think it is worth a trial to prevent such a catastrophe.
"Still, if the Department thinks it better to remain on the defensive, and guard as far as possible all the avenues of approacfh, and await the time of the enemy, I am ready to adopt this course. You have, therefore, only to inform me...."
Sorry, my bad, a mistype.
Longstreet & Lee were not in Carlisle, northwest of Gettysburg.
Longstreet's units were coming into Chambersburg, nearly 30 miles straight west of Gettysburg.
And the key point here is that, on July 1, Longstreet was BEHIND Hill in filing through the narrow pass in South Mountain, on the way east to Gettysburg.
That's why it took him so long to get there.