No chance the world powers (US, China, France, UK, etc.) will allow something like that unless they are the one who create and enforce it. It's basically the first line of defense for homeland security: don't let terrorists into your country.
I doubt the US would ever outsource something like this (border security) to an international organization like INTERPOL. It uses INTERPOL to track down and catch wanted people abroad, but that's it.
What do you think is stopping the US? It could be privacy but I don't think that would hold up in the public court as the US is way worse than EU when it comes to privacy regulations... what else? Control?
And we're not talking about "border security" in its entirety, just the screening/identifying part.
Sure, no problem with that, use Interpol's screening tech as a tool - that's what it is. Just a repository of information that you access and then decide what to do with the result just like any other API hit