[backup-client][native_rust_library] update backup client to run cleanup on any failure

Summary:
ENG-9656.

Previously, we ran cleanup only if the final stage failed, potentially leaving some files in storage - this diff fixes this and attempt cleanup on each failure.

Depends on D13929

Test Plan: N/A

Reviewers: bartek, tomek

Reviewed By: bartek

Subscribers: ashoat

Differential Revision: https://phab.comm.dev/D13930