Correct, even in adversity, Grant is trying only to get at his opponent, as soon as possible, not avoid him. Terrain or politics may be against him, but they do not distract him.
And "as soon as possible" is within the bounds of the conditions which actually prevail. Grant doesn't launch over the Rapidan for the Overland until May because that is when everything was lined up to make it possible, just as one example.
However, in a lot of cases there is a continuum going on. Someone can launch on their campaign and try and attack the enemy at the earliest possible moment by rushing, but it is often the case that a delay to make sure things are actually in order before attacking means that the actual offensive
movement is more likely to succeed.
In the example of Buell getting troops over the Duck, there are basically three possibilities here.
1) The river could have been bridged faster.
2) The river was fordable before the date at which it was actually crossed.
3) Neither of these applies.
It is quite possible that an "utmost effort" could have bridged the river faster, or could have allowed for troops to ford the Duck beforehand; even if neither is the case then it's obviously possible to swim. But we have to go on from there to the question of whether or not doing these faster movements would be (A) worth the trouble and (B) actually result in a better outcome.
The distinction at play for (A) is that, for example, having everyone in the entire army wear themselves out building bridging equipment sixteen hours a day, or hiring in civilian engineers, might get the bridge built faster but would cost more in money and/or unit cohesion. Similarly asking men who aren't experienced swimmers to swim across the flooded Duck river
en masse would get the infantry across, but would probably result in at least a few drownings.
Buell's been ordered to concentrate on Grant, but as far as he's aware Grant will have had the opportunity to fortify his position, and it's not always the case that you must move as fast as humanly possible - it's no good half-killing your men on a forced march and suffering significant straggling losses to get somewhere in three days when there's no need for you to be there before day six.
The distinction at play for (B) is related. Let's say that the day after reaching the Duck river with McCook's division Buell has them swim the river
en masse, and has them march for Savannah.
The resultant formation is moving days earlier, yes. But it's also moving without wagons or artillery (as they're unable to cross the still-flooded river) and so the only supplies it has are the ones which the men can carry with them (usually reckoned at about three days' cooked rations, assuming they survived the swim, and their weapons plus some possibly-wet cartridges). The force has to conduct a move of about 85 miles with the food to cover perhaps thirty under normal march distances - perhaps more like fifty if it's all forced marches without the wagons - but they simply don't have to food to make it, and if they reach Savannah it's without artillery. Meanwhile the bridge is going unbuilt because the men who
would have been working on it are instead somewhere around Summertown, and so the next division has the same basic problem to solve.
I think there is not sufficient preponderance of evidence to conclude that Buell was doing anything other than making a good-faith effort to march troops to Grant as soon as feasible. I think there
is good reason to conclude that his judgement was wrong about the best way to do that, in that sending troops by river probably would have got at least some of them to Grant sooner, but I also think that we don't have sufficient information to know whether the
end result would have been to have more troops able to participate in Shiloh by the end. (As this would require sending four divisions in time, which is a significant lift and there may not have been enough shipping for the round-trips required to carry the lot).
Buell's principal error of judgement would thus be his determination of how long repairing the bridge over the Duck at Columbia would take. If it had taken two or three days, everything would have been fine; I suspect however that for a number of reasons Buell's understanding of how long bridging would take was based on "wrong knowledge" - the Mexican-American War didn't exactly see many ARKstorm events.
Bringing the