Grant's objective in the campaign was clear: hold the Army of Northern Virginia down and batter it to pieces by any means possible.
Not really. What Grant actually wanted to do was get on Lee's lines of supply/
communication and force Lee to attack under unfavorable circumstances. He
came within an ace of this at both Spotsylvania and Petersburg. At the former,
Lee was aided by a forest fire (See Catton,
Grant Takes Command for details),
and at Petersburg, incompetence in the Army of the Potomac's command. He
eventually accomplished this by extending the Petersburg lines far enough west,
causing Lee's March 25, 1865 attack on Fort Stedman. He also was determined to
get to the James River, so although he kept attempting to get behind Lee, he was
force to batter his way to the James.
This he (mostly) successfully accomplished by mauling it in several bloody engagements and finally locking it up inside the Petersburg fortifications; only Early's brief foray marred the accomplishment. But Lee had an objective in the campaign as well, and it was not simply "not to lose": he had to defeat the Federals in some way and keep the Army of the Potomac north of the Rappahannock, whether by offensive action or successful defensive action. In this, Lee failed horribly.
He certainly did succeed in mauling Lee's army. I have to disagree with your
assertion that 'Lee failed horribly' at his objective of 'not to lose'. In the summer
of 1864, this was working quite nicely, and Lincoln himself thought his re-election
chances were close to nil. That the situation turned was not due to any defeat
Lee suffered, but Sherman's capture of Atlanta assisted by Sheridan's victories in
the Shenendoah Valley.
Compare the Overland Campaign to the Seven Days' Battles, a campaign which Lee is usually said to have won by defeating McClellan and the Army of the Potomac. In almost every single engagement in the Seven Days' campaign, the rebels suffered horrible losses, were poorly coordinated, and the only reason they advanced at all was because McClellan was intent on retreating anyway. How is that different from the Federal performance in the Overland Campaign? (Oh, I know how: the Federals took fewer losses in 1862 than did the Confederates in 1864.) Lee gets his army battered but successfully maneuvers his enemy away: he's lauded as a war hero. Grant gets his army battered but gives his enemy a battering of his own and successfully maneuvers him into a corner: he's vilified as an uncaring butcher who only won because he had more bodies to throw into the meat grinder. The hell?
Yup, that sums that part up quite nicely.
Re Shiloh : Yes, Grant did get surprised. BUT, he had stabilized the situation
before
the reinforcements arrived. They made the counterattack possible.
And if Lew Wallace had managed to show up mid afternoon, he would likely have been
able to make a devastating flank attack on the Confederates.
Re Overland Campaign casualties:
I think it's important to keep in mind that most historians are of the opinion that
Grant and Lee suffered around the same percentage of casualties (due to sketchy
Confederate records, it's not possible to be definitive on this one way or the other).
Also, the Army of the Potomac lost thousands of men during this time due to expiration
of enlistments, and did not really have as much of a numbers advantage as most
people believe.
Re: Who They Faced:
One thing in which Lee and Grant were alike was that if an opponent made a mistake
in their front, immediate advantage would be taken of that mistake - Lee happily
taking gift victories from McClellan, Pope,and Hooker, and Grant taking gift
victories from Floyd/Pillow, Pemberton, and Bragg.
Re Sherman:
Sherman was not in the class of Lee or Grant (or Forrest, for that matter). He
was the most outspoken opponent of the plan which succeeded at Vicksburg,
because he couldn't see how it could succeed without a supply line.
He always preferred maneuver over trying to destroy an enemy army (he let
the garrison at Savannah escape, even though he could easily have trapped it,
something Grant would never have done).
And after all this blathering, I take Grant, because I think he was a
superior strategist. I would also take him over any other American general with the possible
exception of Bradley (Eisenhower's role IMO makes a comparison too difficult).