Why can we assume $U_i$ arbitrarily?

Well, you can't always choose a given potential energy arbitrarily, but we can choose a zero of potential energy wherever we like, and it won't change the answers to the problems. The reason is that what matters for energy conservation is change in potential energy, $\Delta U$. If you have an energy conservation equation, say $U_i + K_i = U_f + K_f + E_{\rm th}$, choosing a different zero of potential energy corresponds to adding or subtracting the same thing to/from both sides of the equation-- so the answer will be the same.