What does this PR do?

Fixes the Matrix gateway receive path by starting mautrix's own syncer after handlers are registered. Direct client.sync() calls return sync payloads but do not drive add_event_handler(...) callbacks, which left the adapter connected and able to send while never responding to incoming Matrix messages.

Related Issue

Fixes #7914

Type of Change

• 🐛 Bug fix (non-breaking change that fixes an issue)
• ✨ New feature (non-breaking change that adds functionality)
• 🔒 Security fix
• 📝 Documentation update
• ✅ Tests (adding or improving test coverage)
• ♻️ Refactor (no behavior change)
• 🎯 New skill (bundled or hub)

Changes Made

• gateway/platforms/matrix.py: start client.start(None) after auth, handler registration, initial sync, and initial key sharing; stop the mautrix syncer during disconnect; keep the Hermes background task for joined-room/DM/key-share maintenance instead of direct event-dispatch sync polling.
• tests/gateway/test_matrix.py: update the fake mautrix client, replace stale direct-sync-loop expectations, and add lifecycle coverage for starting/stopping the mautrix syncer.

How to Test

1. python -m pytest tests/gateway/test_matrix.py -q -x --timeout=60 → 152 passed.
2. python -m pytest tests/ -q -x --timeout=60 → stopped at tests/acp/test_approval_isolation.py::TestAcpExecAskGate::test_interactive_env_var_routes_to_callback; this ACP approval-path failure is outside the Matrix gateway files changed by this PR.
3. python -m ruff check gateway/platforms/matrix.py tests/gateway/test_matrix.py → passed.
4. python scripts/check-windows-footguns.py gateway/platforms/matrix.py tests/gateway/test_matrix.py → passed.
5. git diff --check → passed.
6. python -m ruff format --check gateway/platforms/matrix.py tests/gateway/test_matrix.py → reported pre-existing whole-file formatting churn in these Matrix files, so no broad formatter rewrite was applied.

Checklist

Code

• I've read the Contributing Guide
• My commit messages follow Conventional Commits (fix(scope):, feat(scope):, etc.)
• I searched for existing PRs to make sure this isn't a duplicate
• My PR contains only changes related to this fix/feature (no unrelated commits)
• I've run pytest tests/ -q and all tests pass
• I've added tests for my changes (required for bug fixes, strongly encouraged for features)
• I've tested on my platform: macOS on darwin-arm64

Documentation & Housekeeping

• I've updated relevant documentation (README, docs/, docstrings) — or N/A
• I've updated cli-config.yaml.example if I added/changed config keys — or N/A
• I've updated CONTRIBUTING.md or AGENTS.md if I changed architecture or workflows — or N/A
• I've considered cross-platform impact (Windows, macOS) per the compatibility guide — or N/A
• I've updated tool descriptions/schemas if I changed tool behavior — or N/A

Screenshots / Logs

N/A.