And in a way Google is already doing this in person with their interview process: they ask obscure questions aimed at knowlege gained in a computer science degree (or by good test preparation).
It’s not a perfect solution (and I despise it), but it does resolve a lot of those issues.
I doubt the main problem is ensuring integrity, it's that it's hard to test knowledge and capabilities reliably, and that's already a hard problem regardless of where the candidate is.
You need a system that encourages people to optimize for maximum problem solving ability, but a formal grading scheme encourages people to optimize for that instead. This is the single biggest problem with online classes and is why online degrees are not considered as highly as in-person ones.
This is how companies have offices across the world, more and more in low cost locations. It this is what you want ...
The test is merely to help discern how good a job the schools have done.
The etymology of educating is to “lead someone out of” something - namely ignorance.
Universities are about a combination of education and certification but the certification is overwhelmingly what people pay for.