refactor: apply thermo-nuclear quality review fixes

[Microck]

Summary

• Fix search auth fallback state so session-preferred search does not store an API key in the session fallback slot.
• Simplify CLI output-format dispatch by using existing Display implementations and removing template unwrap() branches.
• Harden live Kagi account flows by retrying lens mutation lookups, validating lens names Kagi truncates, and making live Assistant tests use the account's available model catalog.
• Refresh v0.9.0 Homebrew/Scoop packaging references and align release/npm workflow assertions with the current CLI/package behavior.
• Update docs/examples for the live Assistant model slug.

Cleanup

• Removed review-only .workflow/thermo-nuclear-code-quality-review artifacts.
• Removed generated .dora files from the branch.
• Kept the branch scoped to project source, docs, CI, and packaging files.

Verification

• cargo fmt --check
• cargo test -q --locked
• cargo clippy --all-targets --all-features -- -D warnings
• Earlier full verification included release build/package checks, serial ignored live tests, and live CLI smoke against the provided test account.