test(health): router-level coverage for /healthz/ready

[sergey3bv]

Summary

• Added router-level coverage for /healthz/ready when DB is unavailable by mounting the route.

Avoid naming corporate partners, customers, or other sensitive external brands in this PR title or body. Use generic descriptors unless a maintainer explicitly approves the public reference.

Motivation

• A follow-up Improve keycast health probes #93

Related Issue

• Closes test(health): add router-level coverage for /healthz/ready database-unavailable path #135

Testing

• cargo test --workspace --verbose
• cargo clippy --workspace --all-targets --all-features -- -D warnings -A deprecated
• cargo fmt --all -- --check
• Manual verification completed

Visuals

• UI/web change with screenshots/video attached
• No visual change
• Visuals and text avoid sensitive external brand or partner names unless explicitly approved