The US cannot find enough volunteers for its armed forces. On the other hand it has an unlimited supply of volunteers for the right to live in the US. The two do seem to offer a solution. In Roman times one of the routes to citizenship was to serve in the legions. Why can't America adopt the same system?
One possible problem is that the foreign legion could be infiltrated by America's enemies. That can, of course, happen anyway - the fact that someone is an American citizen does not mean they are loyal.
It is unlikely that foreign recruits would simply integrate into the ordinary armed forces. The US armed forces are extremely expensive to run - this is one reason why it would not be easy to simply raise pay rates to boost recruitment. To attract recruits from countries with lower standards of living would be much cheaper.
There are, of course, alternatives. Britain won the Seven Years' War - ironically the war that gave her control of North America - by subsidising allies. Since British soldiers were highly paid it was cheaper to pay Prussia to field an army than to field a British Army.
The Soviet Union used Cuban armies to invade African countries. Subcontracting other countries to fight wars has a long pedigree. The Economist has suggested that the UN should have an army manned by poor countries and financed by rich ones.
But why would America finance an army that could only be used with French, Russian and Chinese permission when it could finance one under its own control?