It is true that the idea of using "cavalry" means to give infantry additional mobility has inspired mechanized warfare. However, Forrest was merely one of many generals throughout history who were known for "mounted infantry" tactics, meaning that they use horses to move, but fight dismounted.

Specifically on US mechanized tactics development, you can read on "Through Mobility We Conquer: The Mechanization of U.S. Cavalry," by George Hofmann, where he writes,

This doctrine was a throwback to how the U.S. Cavalry, both Union and Confederate, generally fought during the Civil War. Embedded in its thinking were the exploits of noted cavalryman, such as Nathan Bedford Forrest, John Hunt Morgan, James Ewell Brown (Jeb) Stuart, Joseph Wheeler, John Buford, Benjamin H. Grierson, Philip Henry Sheridan, and James Harrison Wilson, to mention a few.

However, he further cautioned,

The Civil War tradition, however, collided with new opportunities for warfighting due to the introduction of rapid-firing weapons and new mechanical mounts developed during World War I. The failure the absorb new opportunities regarding advanced in technology was not obvious to the cavalry after the war. It still looked to the past.