The reason there is no draft in the US is not what you think, this is the reason
In the 60’s there was massive anti-war protests in the US, Nixon felt most of the protests was not really anti-war or anti-american, but really anti-draft. Middle Class young people were being forced to go to a war they did not want to go to and would protest.
By making the army all volunteer, only those who really wanted to serve would serve. People generally only protest when it affects them directly . They might burn their own draft card and risk getting arrested, but they are less likely to risk protesting and getting arrested for someone else.
It took about 4 years until Nixon was able to get rid of the draft