[Frontend] Resolve PrivacySettings TypeScript errors without weakening strict mode

## Problem

The settings component contributes multiple type-check failures around form state and API responses.

## Relevant code

`src/components/Settings/PrivacySettings.tsx`

## Proposed scope

Replace unsafe assertions with typed state and response models, retaining strict TypeScript and covering save/error paths.

## Acceptance criteria

- [ ] The reported behavior is reproduced or characterized with a focused automated test before the fix.
- [ ] The implementation satisfies the proposed scope without weakening TypeScript, lint, authorization, privacy, or error handling.
- [ ] Success, empty/loading where applicable, failure, and boundary cases are covered.
- [ ] Existing related tests pass and new regression coverage is included.
- [ ] User-facing behavior remains accessible by keyboard and at mobile viewport sizes where UI is affected.
- [ ] Documentation is updated when the change alters configuration, contracts, security assumptions, or contributor workflow.

## Contributor notes

Base the work on the latest `main` branch. Keep unrelated refactors out of the pull request and include screenshots or recordings for visible changes. Never use real pet, medical, contact, wallet, or credential data in fixtures.

Start by reading src/components/Settings/PrivacySettings.tsx and reproduce or characterize its reported type-check failures with a focused automated test. Trace the form state and API response types through save and error paths, then verify strict TypeScript remains enabled and related tests cover success, empty/loading, failure, and boundary cases.

Written by the indexing model from the issue text.