I guess the point is that when people nondimensionalize Navier Stokes, they neglect to mention they are nondimensionalizing it in a specific geometry with a specific flow (which allows you to consistently define $L$ and $U$). Varying an infinite dimensional parameter like $u_0(x)$ leads to trouble when using the Buckinham-pi theorem, although it's still not clear to me why.

I think I figured it out, the issue is that typically when people nondimensionalize NS they compare two similar scenarios i.e. a certain kind of flow in a certain geometry. Changing the initial condition $u_0(x)$ is too drastic of a change for us to expect it to be modeled by a single parameter ($Re$) family of solutions. How the problem should really be looked at is $u(\rho,\nu, U, L) \rightarrow \tilde{u}(Re)$ in which case 4-3 = 1 and Buckinham pi makes sense.