Now that we have a volunteer army, there are differences between how the wars of today are fought compared to the Viet Nam war. The Viet Nam war had literally an order of magnitude more deaths than our current wars. Having a volunteer force means the soldiers have to treated better and paid more than a draftee. These wars have been a waste of lives and money but things would have been much worse if there were millions of draftees serving in the middle east and Afghanistan. Having a volunteer force in the 1960s would have meant that if the Viet Nam war would have been fought, there would have been with a lot more concern for the loss of life of the solders - it likely wouldn't have been fought anything like it was fought.
The USA should use this opportunity to step back and re-think whether registering for the draft is needed in this age. The whole draft registration program is simply a waste of millions of dollars a year.
I don't (off the top of my head) remember my sources for claims about poverty being a mechanism for manning the US military, so I won't push the point. I'll leave it at saying that I wish you were right.
You're definitely right that it's a recruiting pressure - that and the free college and the path to naturalisation. But that only holds as long as the casualties are relatively low. And the ~4k dead and ~30k wounded of Iraq was relatively low compared to the ~58k dead of Vietnam. It's not clear what kind of non-nuclear war would bring that level of dead again, but I reckon restarting the Korean war would do it.
And that's also why minimum-wage restaurant workers are going on strike in the coronavirus era.