That’s an easy enough problem to solve if we had the appetite to solve it, why couldn’t we legitimize the roles illegal immigrants currently do right now with a Singapore style migrant worker program?

Probably because the average Chinese is far more skilled than your average illegal immigrant.

It's not just raw labor force. It's capable engineers, designers, architects, etc. How is illegal immigration going to solve this?

To be clear, I think that there is an existing, legal migrant worker programme. They do most of the non-automated farm work in Central Valley California. Also, the Singapore system is a bit ugly if you look closely. I hope the US can do better.