Is $X \perp\!\!\!\perp Y$ a conditional independence, arguing that the independence is conditioned on an empty set of random variables? If so, does that mean that an unconditional independence is always a conditional independence (but the converse is false)?
Yes, I would say, $X \perp\!\!\!\perp Y$ can be thought of as conditional independence, but conditioning on the full sample space $\Omega$. The probability of the empty set impossible event is zero, and you cannot really condition on an event with probability zero.
Conditioning on the full sample space really is saying that "the only fact that I know is that some outcome in the sample space happened", as it must if your model is any good.