Grant had one primary political rule, he never criticized the President. The comments about continual pressure, resulting in a victory attrition, if not in any other way, are just paraphrasing for the public, what Lincoln had been trying to get from McClellan and Halleck.
Lincoln and McClellan had discussed early in the conflict, but Halleck and others were unable to implement continual, co-ordinated pressure.
What Grant did was far worse, but much more effective.
The plan was to finish the blockade by capturing all the harbors and cities that had a railroad connection. He also was breaking down the railroad system of the Confederacy, which would make it impossible for food and forage to reach the armies, and also impossible to reach the few remaining cities of the Confederacy.
1. Farragut captured Mobile Bay.
2 Grant severed the railroad connection between Richmond and Weldon, NC.
3. Sherman captured the railroad town of Atlanta, GA. He did not even permanently occupy it. It was of very little value to the US.
4. Sheridan confiscated or destroyed everything in the Shenandoah Valley located close enough to a railroad to be valuable to the Confederate army in Richmond.
5. Sherman marched through Georgia and captured Savannah as intact city. He let the Confederate garrison flee the city, because he did not want a siege. He wanted the port, which was useful to the US.
6. Butler failed to hold his position north of Fort Fisher, and he was dismissed. Terry and Porter completed the operation, and Schofield was brought around from Tennessee and landed at Wilmington.
7. Grant severed the Southside Railroad connection to Richmond, and Lee abandoned Richmond and retreated mostly along the line of his last remaining railroad.
Grant was engaged in war against not just the army of the Confederacy. Sherman and Sheridan expressed it more directly. But Grant hid the hard truth that he abandoned Lincoln's preferences, and Grant pursued his own plan for a 6 state siege.
Thank you yes, it is clear from any level of familiarity of Grant's Campaigns as Lieutenant-General, that he targeted Confederate infrastructure and supplies. He said once that it was 'more important to target armies than cities,' or something like that, but clearly, the Overland Campaign failing, he took full liberty to tear of railroads, burn the Valley, commit to Sherman's march to the Sea, attack and lay siege to Petersburg, Lee's advanced supply depot, and possibly other examples I'm forgetting and that you may have mentioned.