Make `OverlaySdkOnLKG` and cross-architecture overlays incremental

Part of #48070. Split `OverlaySdkOnLKG` preparation from execution so cross-architecture follow-up targets retain their item inputs when overlay copying skips. Own stale test-host outputs, preserve stage-0 and bundled-version behavior, and avoid repeated enumeration and copying where inputs are unchanged.

**Depends on:** #55871 and #55872.

Start by reading the related work in #55871 and #55872, then trace the OverlaySdkOnLKG preparation and execution paths. Done means cross-architecture follow-up targets retain item inputs, stale test-host outputs are handled, stage-0 and bundled-version behavior remains intact, and unchanged inputs avoid repeated enumeration and copying.

Written by the indexing model from the issue text.