Record one funnel step from the /start/ page.

api_auth.PUBLIC_PATHS has listed this route since the funnel shipped and marketing_scoreboard.py reads the file it writes, but the route itself was never implemented — so /start/'s beacon POSTed into a 404 and funnel_events.jsonl stopped growing on 2026-07-05. Every visit, email and account event since then was lost, which is why the funnel could not be measured at all.

Appends {ts, session, event, arm, path} — the exact shape already in the file. Never raises: a beacon that 500s would be no better than a 404.