Lets also remember that Grant was relieved of his command shortly after the Battle of Shiloh. If it hadn't been for his political connection with Elihu Washburne (Lincoln's close personnal friend) because of Grant's father Jessie Grant political connection with Washburne, Lincoln, Henry Halleck and the others of the Gelena, Illinois gang, he would have been canned and not been in command of the AoP.
The same for Sheridan, who was also canned and got a rare second chance, because of his friendship with Grant. How many other Union Generals got that same "second chance" that both Grant and Sheridan recieved?
It was actually Chattanooga and Grant's handling the relief of that place and Rosecrans army and Missionary Ridge that propelled Grant to the command of the AoP. It was Henry Halleck, Grants old friend and another member of the Gelena Gang, who sent Grant to take command of the situation at Chattanooga. Grant had been sitting at Vicksburg with nothing to do for several weeks.
It was Halleck and Wasburne who championed Grant to Lincoln for Command of the AoP. after Missionary Ridge and the near destruction of Bragg's Army of Tennessee. But even then after recieving command of the AoP, Grant did not win a straight up battle against Lee during all of 1864. Lee held every battlefield that they fought on. Even with the massive strenght of the AoP under Grants command he never broke Lees ragged thin lines until April 1865. Any one of those battles losses would normally have cost Grant his job had he been treated and his ability judged in the same manner as the previous commanders (McDowel, Pope, McCellan, Hooker, Burnside, Meade) of the AoP.
While there is no doubt that Grants military tactics and fighting style were straight forward and ahead of their times, (think World War I and the trench warfare of France 1917) there is no doubt that he also lead a very charmed, politically connected and protected, career.