WW2 was probably the closest we ever came (over 63% draftees). It was America's brief and shining moment where we ,across all class lines, came together as a people and fought something very evil.
But you're right about the rich & privileged finding ways to weasel-out of the fighting. It was even happening during the American Civil War on the Union side.
This is from Wikipedia:
"The New York draft riots (July 1316, 1863), known at the time as Draft Week, were violent disturbances in Lower Manhattan, widely regarded as the culmination of working-class discontent with new laws passed by Congress that year to draft men to fight in the ongoing American Civil War. The riots remain the largest civil and racial insurrection in American history, aside from the Civil War itself.
U.S. President Abraham Lincoln diverted several regiments of militia and volunteer troops from following up after the Battle of Gettysburg to control the city. The rioters were overwhelmingly working-class men, who resented that wealthier men, who could afford to pay a $300 (equivalent to $9,157 in 2017) commutation fee to hire a substitute, were spared from the draft."
That was an eye-opener for me.