If I recall correctly, Lee had a hand in this situation as well.
Very true. I didn't know much about this situation, so out of curiosity read the relevant correspondence in the OR, and it's pretty obvious that Lee had a major hand in this fiasco.
It is in Series 1, Volume 36, part 3...
On June 3rd, Hancock writes there are many wounded men between the lines, but that they will be "brought off at night."
On June 5th, at 1:00 pm, Hancock writes Meades adjutant that there are still wounded in front of Barlow and asks if they could be brought off. Meade writes Grant asking if it would be possible to "ask, under a flag of truce, to remove the wounded now laying between our lines?" This is the first time Grant is informed there are still some wounded on the field, and he grants permission to "send a flag proposing to suspend firing until each party get their own" wounded.
A half-hour later, Meade writes to Grant that the confederates won't accept a communication under a flag of truce unless it comes from Grant. Apparently the confederates won't honor Meades authority for some reason.
Grant sends a communication to Lee immediately, proposing that unarmed men with litters be allowed to pick up the wounded when no battle is raging. Lee writes back the same day (the 5th) that he fears there might be difficulties or misunderstandings unless it is done in the usual way under a flag of truce.
On the 6th, Grant receives the message from Lee dated the 5th, and writes back to Lee proposing Noon to 3:00 pm for his men to collect wounded under a white flag.
Lee writes back refusing consent to Grants proposal, saying that "either party" must ask for a truce in the "usual way" in order to pick up prisoners. So Grant writes back and accedes to Lees demands, and asks for a "suspension of hostilities" to satisfy Lee, and asks that Lee designate two hours for Grant to pick up his wounded.
So in all, roughly 24 hours has passed, from the time Grant was told about the wounded on the 5th, until the time on the 6th that he asked Lee for a general suspension of hostilities in order to collect the wounded. They each exchanged two communications, and the slowness of the delivery is really what delayed the resolution.
At this point, the whole thing turns into a tragic comedy of errors, with messages between Grant and Lee not being received until
after the times requested for the truce, and a confederate litter party mistakenly being captured by union troops. The confederates thought they were in the middle of a truce, but it had not yet been communicated to the union side.
It was, no doubt, a fiasco. But not all on Grant.