Not actually true.
Lincoln offered him a major generalship with responsibility for the defense of DC.
Lee considered it, and tried to get assurances through Francis Blair (a family friend whose son was in the Lincoln Cabinet) that his role could be purely defensive without having any obligation to invade Virginia.
When Blair told him there was no way anything like that could be guaranteed, he turned down the commission.
He was a well-regarded officer, but neither Lincoln nor Davis saw him as the truly great commander he would become, until he had already become such a general.
Correct. His first command was in West Virginia and the confederates were pretty easily routed. Davis then sent Lee to build coastal defenses in Georgia and South Carolina. After that, he was assigned as a military aid to Jeff Davis and only after Joe Johnston was wounded at Seven Pines a full year into the war did Lee replace him as commander of the Army of Northern Virginia, a move that was criticized at the time by the Confederate press who called him "Granny Lee."