Autocorrelation of Addition of Two Independent Signals
You're correct as the Cross Correlation function vanishes.
This has the implicit assumption the process has zero mean (Actually, at least one of them).

Namely, in order to have $ {R}_{XY} \left( \tau \right) = 0 $ having $ X \left( t \right) \perp Y \left( t \right) $ isn't enough but at least of them has zero mean (Namely, $ \mathbb{E} \left[ X \left( t \right) \right] = 0 $, $ \mathbb{E} \left[ Y \left( t \right) \right] = 0 $ or both).

Pay attention that in practice for a single realization the empirical cross correlation won't vanish.