docs: update README API section

[Si40Code]

Summary

• Update the root README API section now that the public Developer API docs are available.
• Remove beta/draft API wording and link to the quickstart, authentication, calls, and API reference docs.
• Add API key dashboard guidance, production base URL setup, batch create-call curl example, and read-result example showing structured results, outcome fields, recipient results, and transcript turns.

Validation

• node scripts/check-branch-name.mjs --branch docs/readme-api-section
• git diff --check
• node scripts/check-manifest-versions.mjs
• node scripts/sync-install-doc-versions.mjs --check

Note: full pnpm checks were not run locally because the installed pnpm is 8.10.2, while this repository requires >=10.18.3; corepack pnpm previously failed with a package-manager signature key error before running project checks.

Release decision

No release needed. Root README documentation only; no package behavior, install flow, or published artifact changes.