While out-of-time-order correlators establish a causal light cone for operator spreading, they do not guarantee that the parameter sensitivity carried by the operator remains locally recoverable. We examine the distinction between operator spreading and metrological recoverability for a parameter encoded in a single site of an XX spin chain subjected to a $U(1)$-breaking transverse field. We evaluate three levels of local metrological accessibility: the bare single-site quantum Fisher information (QFI), the QFI recovered by a variational sweep decoder acting on a finite spatial block, and the exact block QFI. In the integrable limit, the sensitivity propagates as a one-magnon wave packet, and a block decoder with a single output qubit recovers the full block QFI. Breaking magnon-number conservation couples the parameter tangent state to multi-magnon sectors. We analytically demonstrate that the local QFI has no first-order correction in field strength; the leading enters at $𝒪(h^2)$, as required by an exact parity symmetry, and is consistent with the scattering into the two-magnon sector. At stronger coupling, the decoded QFI falls below the exact block QFI. The time-averaged block QFI itself falls below its integrable-limit value, showing that part of the sensitivity has left the block. Comparing these quantities separates the arrival of operator support from the local accessibility of metrological information.