Grant was just implementing Lincoln’s policy of charge first and ask questions later.
The only other commander who would've done a better job than Grant was George Thomas. He was a Virginian who stayed loyal to the Union, so he didn't have the political backing that Grant and Sherman did (from Ohio). He was also considered "slow" because he was slow moving- from an accident he had falling off a horse.
However, once he commenced an operation, his armies were fast and devastating. His only two operations in which he was in command, Mill Springs, KY and Nashville, resulted in annhilation of Confederate forces opposing him and a complete Union victory.
https://www.battlefields.org/learn/biographies/george-thomas