feat: add API liveness check to /health endpoint

[nickmeinhold]

Context

During the Mar 17 outage, claude-sonnet-4-6 stopped being accepted on the Max plan OAuth endpoint. The /health endpoint reported healthy the entire time because it only checks bot polling, MCP server connectivity, and OAuth token expiry — not whether the Claude API actually accepts requests.

Proposal

Add a synthetic API call to the health check — a cheap max_tokens=1 request that verifies the model is reachable:

// In health.ts, add to component checks:
try {
  await anthropic.messages.create({
    model: MODEL,
    max_tokens: 1,
    messages: [{ role: "user", content: "ping" }],
  });
  components.api = { status: "up" };
} catch (err) {
  components.api = { status: "down", error: err.message };
}

Cache the result for ~5 minutes to avoid excessive API calls on repeated health checks.

Acceptance Criteria

• /health includes an api component with up/down status
• API check is cached (not called on every health request)
• Overall status degrades to degraded when API is down