I don’t think you can do it from a TN anyway, so you’d need to switch to an immigrant intent visa (H1-B, EB, etc.)
You actually can, a lot of people do that. You just have to time it so that it completes before your TN renews, because after you start the GC process, you will not be able to renew TN again (since you violated non-immigrant intent).
It only really works for Canadian born people, because it'll take 1-1.5 years for GC, Mexicans will take longer so they can't really do that on TN. And as you said, India/China wait list is around 60 years now I believe.