Build: Migrate @storybook/api to strict types

[stilt0n]

Part of #22176

What I did

Change tsconfig.json of @storybook/api to strict type-checking for completeness' sake. Doesn't really do much because @storybook/api is just re-exporting @storybook/manager-api.

How to test

See if CI type check task runs clean

Checklist

• Make sure your changes are tested (stories and/or unit, integration, or end-to-end tests)
• Make sure to add/update documentation regarding your changes
• If you are deprecating/removing a feature, make sure to update
  MIGRATION.MD

Maintainers

• When this PR is ready for testing, make sure to add ci:normal, ci:merged or ci:daily GH label to it to run a specific set of sandboxes. The particular set of sandboxes can be found in code/lib/cli/src/sandbox-templates.ts
• Make sure this PR contains one of the labels below.

["cleanup", "BREAKING CHANGE", "feature request", "bug", "build", "documentation", "maintenance", "dependencies", "other"]

[ndelangen]

Thank you for your time and effort @stilt0n

This package has been deprecated and removed in 8.0.
I'm sorry but that means I can't merge your PR.