Prod web via launcher: handle missing clients/web/dist and verify end-to-end in CI

[cliffhall]

Carved out from the "did not solve" list of #1452 (port of CLI/TUI/launcher to v2).

Problem

mcp-inspector --web (production, no --dev) serves static assets from clients/web/dist/, which only exists after npm run build. If dist/ is absent, the prod web path serves a broken page (serveIndexHtml ENOENT on index.html). The launcher smoke test only checks --help / --cli --help / --tui --help; it never starts prod web end-to-end, so this path is unverified in CI.

Proposed solution

Pick one (or both):

• (a) Have the launcher detect a missing clients/web/dist/ and either build it on demand or exit with a clear, actionable message.
• (b) Add a CI step that builds web and starts the prod web launcher path end-to-end, asserting / returns the SPA (HTTP 200) with the injected token.

Acceptance criteria

• mcp-inspector --web either works after a documented build step or fails with a clear, actionable error when dist/ is missing (no silent broken page).
• CI exercises the prod web launcher path end-to-end against a built dist/ (serves index.html 200).
• Behavior documented in AGENTS.md / launcher README.