Let the page check target a deployed URL

[alexandrefimov]

Summary

bench/check_page.py hardcoded the local server URL, so the same gate CI runs
against a fresh build could not be pointed at the live site. It now reads
QUERY_DOCTOR_PAGE_URL and falls back to the local build.

Used it to verify the deployment from #18 end to end against
https://alexandrefimov.github.io/Query-Doctor/ : boot 1898 ms on a cold cache,
103 KiB sample profile analyzed in 158 ms, 8 requests, no external hosts, no JS
errors.

Type

• Tests only
• Documentation

Safety Checklist

• No raw production data, paths, hostnames, or secrets introduced.
• No product behavior changes; this touches a check harness and its docs.

Validation

• ruff check web/bench
• Gate run against the deployed site: pass
• Gate run against a local build: pass

Branch Hygiene

• Targets a review branch, based on current main.

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Assisted-by: Claude Opus 5 via Claude Code