Consequences of Unique Games being a NPI problem
To address your revised question:


  If UGC is false, is it still NP-hard to give a $(2-\epsilon)$ polytime approximation algorithm for Vertex Cover?


If it were known that it is NP-hard to approximate VC better than a factor of 2, even if the UGC is false, then we would unconditionally know that approximating VC to a factor better than 2 is NP-hard. This also works with a factor smaller than 2. If it were known that if UGC is false, then VC is NP-hard to approximate better than a factor of (say) 1.5, then we would know that it is (unconditionally) NP-hard to approximate it better than a factor of 1.5.

Since we don't know that, I guess the best we can say if the UGC is false is that it's NP-hard to approximate it to within a factor of 1.3606 by the result of Dinur and Safra (2005). (I think that's the current best-known inapproximability result.)