Fold the DB-ping health check in; delete the watchdog that never ran

scripts/health-watchdog.sh was written after the 2026-08-04 incident, where a
liveness stub stayed green for three days while every database-touching request
path was wedged. Its header says "Meant to run from cron every few minutes". It
was never scheduled — not in any unit, not in any crontab — so it has been dead
code since August, offering coverage it did not provide.

Its one real idea is worth keeping: /health pings the database and returns 503
when that fails, which the gateway route checks cannot see, because `/` answers
303 to the login page without going near the database.

check_app_health now does that poll (two attempts, five seconds apart, so a
single blip is not an outage), verify-deploy asserts it, and health-check
recovers from it by restarting the main service — the action the old watchdog
would have taken. The dead script is removed rather than left to imply cover.

Tests now 11, including a stub server returning 503 to prove a listening-but-
wedged app is caught.

Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_018AwnALZL6bD4VBtATDVv3B

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