It is a mistake to think of the Battle of the Wilderness as a Confederate Victory. While it is true that Union losses exceeded Confederate losses, Grant could replenish his and Lee couldn't. Lee had to destroy the Union army to win, and Grant just had to wear Lee down.
Further, though Grant vacated the battlefield following the battle, he advanced rather than retreating. Lee then had to abandon the battlefield as well to cover Richmond.
Many authors have described the tremendous boost to Union morale that ensued from Grant's actions after the battle. Here is one account (from Wikipedia, because it is on-line and convenient):
Both flanks had been badly bruised, and [Grant's] 17,500 casualties in two days exceeded the Confederate total by at least 7,000. Under such circumstances previous Union commanders in Virginia had withdrawn behind the nearest river. Men in the ranks expected the same thing to happen again. But Grant had told Lincoln "whatever happens, there will be no turning back."
While the armies skirmished warily on May 7, Grant prepared to march around Lee's right during the night to seize the crossroads village of Spotsylvania a dozen miles to the south. If successful, this move would place the Union army closer to Richmond than the enemy and force Lee to fight or retreat. All day Union supply wagons and the reserve artillery moved to the rear, confirming the soldiers' weary expectation of retreat. After dark the blue divisions pulled out one by one.
But instead of heading north, they turned south. A mental sunburst brightened their minds. It was not another "Chancellorsville ... another skedaddle" after all. "Our spirits rose," recalled one veteran who remembered this moment as a turning point in the war. Despite the terrors of the past three days and those to come, "we marched free. The men began to sing." For the first time in a Virginia campaign the Army of the Potomac stayed on the offensive after its initial battle
— James M. McPherson, Battle Cry of Freedom
By another account that I have read, but cannot locate just now, Grant himself stood all night long at the key crossroads, in a dark Union overcoat and smoking a pipe, directing all the traffic South instead of North.
The true victor of a battle is the general who retains the strategic initiative after the battle. All secondary indications of victory, such as casualty figures, prisoners and captured munitions, can authoritatively be summarized in this simple assessment. The whole point of war, and of battle, is to impose one's will on the enemy, and ultimately to leverage this into a destruction of the enemy's resistance to this imposition.
Grant achieved this in The Wilderness in spades, casualty figures be damned.
Further, it is important to recognize that while Washington was the capitol of the North, Richmond was the heart and soul of the Confederacy in many ways, as well as its capitol. The loss of Washington, like the loss of Moscow in 1812, would have been a handicap but not an amputation of the North's ability to continue the war.
Contrariwise, the South could only win by retaining its territorial integrity long enough to exhaust the Union's will to fight. With each loss of territory, the South was incrementally shaved of its already tenuous ability to maintain an army in the field, and thus to continue fighting the war.
Update (just for Lennart Regebro):
A quick comparison on Google Earth shows that the Potomac River at Harper's Ferry is roughly the same width as the Danube channels from Lobau Island to Aspern-Essling; The scale on Google Earth suggests about 400-450 feet, but I have reason to believe that may actually be about double the real figure. The Potomac does get narrower near Brunswick, by about 1/3, but from the appearance also faster and deeper.
The Rappahannock is perhaps just half the width of the Potomac, but that is still a non-trivial obstacle.
Given the difficulty of forcing a river crossing against determined opposition, it seems natural that a both sides would attempt stealth and maneuver as means of achieving a crossing.
Also, a quick glance at a map (such as this or many others available on the web) shows that Fredericksburg and Chancellorsville are far from being western routes to Richmond. In fact, unless one wishes to paddle down the length of a multi-mile wide Potomac due south of Washington, it would be difficult to find any major city on the Rappahannock remotely closer to a direct bee-line between Washington and Richmond to name a battle after.
Given this, one could easily argue that Grant choose the Wilderness route because Meade, Hooker, McClennan and Scott (did I miss any) had amply demonstrated the foolishness of attempting anything more direct.