The Numbers Argument
This is the problem that the core concept of the Overland campaign (as presented, though arguably not correctly) was to use attrition to weaken the Confederate army.
Since we have the CEVs of all the battles, we can conduct analysis with the troops who would actually be there if the combat were conducted in 1862. For this analysis I am assuming the following army sizes:
1864 period armies are as per Wikipedia, to calculate the original CEVs.
The 1862 Confederate Army in northern Virginia is about 70,000 effectives and is fighting from the Wilderness onwards.
Confederate reinforcements arrive after the fighting at Spotsylvania but before Cold Harbor, and represent:
12,000 from the Peninsula and Norfolk (leaving about 12,000 between the two positions)
8,000 from North Carolina (essentially the core of AP Hill's division)
Loring's force in the dept. of SE Virginia replaces Jackson's force detached to the Valley, so doesn't change the numbers.
The Union army starts at 120,000 (1st-4th corps inclusive) but does not get any reinforcements unless noted - this still leaves the defenders of Washington considerably weaker than they were historically allowed to get at any point in 1862 so it seems appropriate.
Run 1 - assuming each battle has the fighting continue until the Union has suffered the historical number of casualties.
Battle of the Wilderness
Historical: 124,500 Union PFD vs. 62,500 Confederate Effectives
Casualties: 17,500 Union, 11,000 Confederate
Alternate: 120,000 Union vs. 70,000 Confederate
Casualties: 17,500 Union, 8,100 Confederate
Battle of Spotsylvania
Historical: 105,000 Union, 52,000 Confederate
Casualties: 18,400 Union, 12,700 Confederate
Alternate: 102,500 Union, 61,900 Confederate
Casualties: 18,400 Union, 8,500 Confederate
Battle of Cold Harbor
Historical: 113,000 Union, 60,000 Confederate
Casualties: 12,700 Union, 5,300 Confederate
Alternate: 94,900 Union, 73,400 Confederate
Casualties: 12,700 Union, 2,500 Confederate
Campaign conclusion: 82,200 Union, 70,900 Confederate
Historical conclusion: 110,000 Union, 55,000 Confederate
The campaign has bled the Union army white, and bringing in 5th corps (complete) afterwards would leave it at the historical conclusion but would leave Washington badly exposed - there'd be the forces in the forts and nothing else. Given the size of the force Lee could historically detach in Early's raid (ca. 10,000), here he could threaten Washington much more effectively.
Run 2: Ditto run 1, but Confederate casualties fixed
Historical: 124,500 Union PFD vs. 62,500 Confederate Effectives
Casualties: 17,500 Union, 11,000 Confederate
Alternate: 120,000 Union vs. 70,000 Confederate
Casualties: 23,600 Union, 11,000 Confederate
Battle of Spotsylvania
Historical: 105,000 Union, 52,000 Confederate
Casualties: 18,400 Union, 12,700 Confederate
Alternate: 96,400 Union, 59,000 Confederate
Casualties: 28,100 Union, 12,700 Confederate
Battle of Cold Harbor
Historical: 113,000 Union, 60,000 Confederate
Casualties: 12,700 Union, 5,300 Confederate
Alternate: 68,300 Union, 66,300 Confederate
Casualties: 42,400 Union, 5,300 Confederate
Functionally this is the destruction of the Army of the Potomac.
Run 3
For run 3, both sides throw in all their possible reinforcements for Spotsylvania - thus the Union gets 30,000 reinforcements after the Wilderness, and the Confederates get 30,000 after the same (this is their pulling in four brigades from the Peninsula and Norfolk, four from North Carolina and two from Loring, with no Valley force). Here using attacker casualty rates as fixed, taking the Confederates as the attackers at the Wilderness, to reflect when the attacker can no longer attack.
Wilderness
Historical: 124,500 Union PFD vs. 62,500 Confederate Effectives
Casualties: 17,500 Union, 11,000 Confederate
Confederate casualty rate: 1 in 5.68
Alternate: 120,000 Union vs. 70,000 Confederate
Casualties: 26,400 Union, 12,300 Confederate
Armies after this battle: 93,600 Union, 57,700 Confederate
Union gets 30,000 reinforcements (5th Corps)
Confederates get 30,000 reinforcements (Peninsula, NC etc.)
Armies now: 123,600 Union, 87,700 Confederate
Battle of Spotsylvania
Historical: 105,000 Union, 52,000 Confederate
Casualties: 18,400 Union, 12,700 Confederate
Union casualty rate: 1 in 5.7
Alternate: 123,600 Union, 87,700 Confederate
Casualties: 21,600 Union, 7,300 Confederate
Armies after this battle: 102,000 Union, 80,400 Confederate
Battle of Cold Harbor
Historical: 113,000 Union, 60,000 Confederate
Casualties: 12,700 Union, 5,300 Confederate
Union casualties: 1 in 8.9
Alternate: 102,000 Union, 80,400 Confederate
Casualties: 11,500 Union, 2,200 Confederate
Campaign conclusion: 90,500 Union, 78,200 Confederate
Historical conclusion: 110,000 Union, 55,000 Confederate
With 20,000 less Union troops than historically present on the James, Lee can keep the Union army in check with about the same number of men he used historically and detach about 20,000 men more to threaten Washington (from his larger army). This leaves Washington's garrison threatened by an at-least-equal force of Confederates and could quite feasibly lead to the loss of Washington.
Overall conclusion: put simply, the Union's pockets are less deep in manpower terms while the Confederate ones are deeper. The Overland campaign's fearsome cost in attrition cannot be sustained with the forces the Union has available relative to the Confederacy.