Q: Huygens' principle or finite speed of propagation?
It's the same thing, Huygens principle is a statement of causality, which means finite speed of wave front propagation.
Without Huygens principle you might imagine that the right-hand-side of the wave equation is a source term that allows a nonzero $v(x,t)$ to appear at $|x|>t+C$, without any relationship to the wavefronts at earlier times. The notion that a nonzero $v$ at some later time is causally related to a wavefront at an earlier time is the content of Huygens principle.
Note added in response to comment: the OP asks about $\mathbb{R}^3$, where all wavelets propagate with a single velocity $v\equiv 1$ and Huygens applies. In even dimensions all velocities $\leq v$ contribute. The support statement in the OP still holds, even though Huygens does not.