Could the Union have won on the same terms in the same amount of time without them?

To answer your last question first, absolutely not. Ulysses Grant and William T. Sherman were vital in bringing the war to a conclusion, and their campaigns shaped the very fabric of the post-War world. In fact, their legacies are still felt to this very day. If you dont believe me, as a Georgian what they think of Sherman.

But were their tactics necessary for the Union to win?

To your second question, Im very much reminded of a quote from Sherman, written in early 1864 on the eve of his Atlanta campaign. He said:

War is the remedy our enemies have chosen. Other simple remedies were within their choice. You know it and they know it, but they wanted war, and I say let us give them all they want; not a word of argument, not a sign of let up, no cave in till we are whipped or they are.

This single quote, I think, really encapsulates the attitudes of Grant, Sherman, Lincoln, and other contemporary pro-Union politicians and generals. Ulysses Grant specifically felt that the great flaw of the Union's early generals were that they were unwilling to fully commit themselves to victory. George McClellan, Ambrose Burnside, and even Henry Halleck all failed to follow up their victories, exploit the Confederates weaknesses, and maximize the Union's strength. Rather than wage a campaign against the entirety of the south, commanders, especially in the east, preferred to box with their adversaries. Rather than strike a decisive blow, many simply spent lives to accomplish nothing. After all, the Army of the Potomac was no closer to Richmond than it had been in 1861, after four years of fighting and hundreds of thousands of deaths.

Likewise, on the economic front the Union had opted for a strategy of slow starvation, but generals were slow to target Confederate industry. Again, rather than strike decisive blows, the Confederates were allowed to extract the maximum amount of men and materiel from their territory. This equipment and these men were used to bolster the Confederate field armies and prolong the war.

The reason, I would argue, why the Civil War ended when it did, and in the way it did, was the Ulysses Grant fundamentally changed the way that Americans conceptualized battles, campaigns, and their strategic value. Prior to the Civil War, generals were trained to fight campaigns of maneuver, with the goal of seizing territory from the enemy. Grant had little interest for territory. Rather, he was interested in the enemy armies and their will to resist. Whatever he did, his ultimate goal was to destroy the enemy army. At every turn, he sought to bring his enemies to battle and destroy them. Failing that, he at least denied them areas of vital interest, industry which sustained field armies, and even the food which they marched off of.

Grant's innovation was that he figured out how to return decisive battles to a hitherto indecisive conflict. Grant moved the Union's armies forward, attacked the Confederates where they were, and attacked them in every location possible whenever it was possible. Grant was able to achieve in one year what his predecessors had failed to do in three.

But its also important to note that this strategy wasnt supported by every person within the Union and without. At the time, many Democrats opposed the war and supported some kind of negotiated peace. George McClellan ran in 1864 on a platform of a negotiated peace. However, the South was categorically unwilling to offer terms acceptable to Lincoln, who would only accept reunification. Like the Union, as the death toll increased, Southerners increasingly expected more from the war. What perhaps started out as a political protest had slowly escalated to a war for national existence. So many in the South, and many in the North, understood that there were only two kinds of peaceful settlements. One which destroyed the North as it existed pre-War or one which destroyed the Confederacy as it then existed.

Sources:

The American Way of War, Russell Weigley

A Great Civil War, Russell Weigley

And Keep Moving On: The Virginia Campaign, May-June 1864 Mark Grimsley