Converts the README from manually-maintained code snippets to using excerpts from a code file.

Also updates the snippet slightly:

• Adds an example of a nullable method, to show that not all are non-nullable.
• Removes the demonstrating of getting the string from the Directory, since that's not related to this package.

Includes sanity check tests of the snippet code to make sure it runs successfully.

Part of flutter/flutter#102679

Pre-launch Checklist

• I read the Contributor Guide and followed the process outlined there for submitting PRs.
• I read the Tree Hygiene wiki page, which explains my responsibilities.
• I read and followed the relevant style guides and ran the auto-formatter. (Unlike the flutter/flutter repo, the flutter/packages repo does use dart format.)
• I signed the CLA.
• The title of the PR starts with the name of the package surrounded by square brackets, e.g. [shared_preferences]
• I listed at least one issue that this PR fixes in the description above.
• I updated pubspec.yaml with an appropriate new version according to the pub versioning philosophy, or this PR is exempt from version changes.
• I updated CHANGELOG.md to add a description of the change, following repository CHANGELOG style.
• I updated/added relevant documentation (doc comments with ///).
• I added new tests to check the change I am making, or this PR is test-exempt.
• All existing and new tests are passing.