You can find all the correspondence online in the Papers of US Grant. Yes, Grant explained that all the armies were to work towards a common center, and eventually he expected Meade would link up with Butler, and he envisioned needing the York or the James at some point for supplies, but it was definitely the plan that Lee's army was the objective point for the Army of the Potomac.
So he said.
So why did he tell Butler he was going to join him on D+10 to besiege Richmond per Butler's autobiography), and that the AoP would come to the Bermuda Hundred:
"Richmond is to be your objective point, and that there is to be cooperation between your force and the Army of the Potomac must be your guide. This indicates the necessity of your holding close to the south bank of the James River as you advance.
Then should the enemy be forced into his intrenchments in Richmond, the Army of the Potomac would follow, and by means of transports the two armies would become a unit. All the minor details of your advance are left entirely to your direction." - Grant to Butler, 2nd April
"You also understand that with the forces here I shall aim to fight Lee between here and Richmond if he will stand. Should Lee, however, fall back into Richmond,
I will follow up and make a junction with your Army on the James River. Could I be certain that you will be able to invest Richmond on the south side, so as to have your left resting on the James, above the city, I would form the junction there." - Grant to Butler, 19th April
"General Butler will operate on the south side of James River, Richmond being his objective point. I will move against Lee's army, attempting to turn him by one flank or the other. Should Lee fall back within his fortifications at Richmond, either before or after giving battle,
I will form a junction with Butler, and the two forces will draw supplies from the James River. My own notions about our line of march are entirely made up, but as circumstances beyond my control may change them,
I will only state that my effort will be to bring Butler's and Meade's forces together." - Grant to Halleck, 29th April
Why did he write to Halleck on 8th May:
"It is not demonstrated what the enemy will do, but the best of feeling prevails in this army, and I feel at present no apprehension for the result.
My efforts will be to form a junction with General Butler as early as possible, and be prepared to meet any enemy interposing. The result of the three days fighting at the Old Wilderness was decidedly in our favor. The enemy having a strongly intrenched position to fall back on when hard pressed, and the extensive train we have to cover, rendered it impossible to inflict the heavy blow on Lee's Army I had hoped.
My exact route to the James river. I have not yet definitely marked out."
"A DESPATCH from Gen. Grant has just been received, —
he is on the march with his whole army to form a junction with you, but had not determined his route." Stanton to Butler, 9th May
Grant, after the Wilderness, did not follow Lee, and did not make Lee's Army his objective point. He was endeavoring to march to the James to be transported to the Bermuda Hundred, in order to besiege Richmond. Lee kept interpositing and forcing Grant to fight. Grant seems to have been living in a fantasy world where Lee was a passive target, and would allow him to do whatever he liked. Unfortunately for Grant, Lee was no Pemberton or Bragg...
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A Further Point ---
Grant's int when he made the proclamations about following Lee was that Longstreet was staying in East Tennessee. Indeed, it was the 7th April before Longstreet was ordered back to VA, and it was the 20th April before a message was sent to Grant telling him that Longstreet had gone.
Grant had wanted to use Burnside's 9th Corps overland (which he gives as 25,000 strong), despite Lincoln having a pet project for it - Fort Fisher. Grant used the int of Longstreet coming east to lever the 9th Corps from this. However, had 9th Corps gone to the Carolina Coast, it would have forced Hoke's and Pickett's divisions to remain there.
Grant's planning is to take ca. 150,000 men overland, against opposition of only ca. 50,000 (<40,000 infantry). He thinks he'll knock Lee out of the way, link up with Butler's 33,000 on the James and besiege Richmond in mid-May.