Fix health readiness semantics

[sylvesterdamgaard]

What changed

• Run the first health check immediately after initial_delay instead of waiting for the first full period.
• Mark supervisor readiness only after a real successful probe via LastCheckSucceeded.
• Feed actual health snapshots into the file-based readiness manager.
• Start the readiness monitor from serve and remove the readiness file during shutdown.
• Add focused tests for immediate first checks and readiness requiring a successful probe.

Why

Dependencies and Kubernetes readiness could be delayed by initial_delay + period, and a transient failed readiness probe could still mark a dependency ready because liveness stayed optimistic until the failure threshold.

Validation

• git diff --check
• go test ./internal/process could not be run locally because go is not installed in this environment.